Generic answer to initial question:

When using quartiles, no knowledge about the underlying data distribution has to be assumed.

When interpreting odds ratio you are actually talking about ratio's of probabilities for 2 mutually exclusive outcomes. Here too the underlying data distribution does not necessarily come into play.

Whenever you want to infer one from the other [quartiles from odds or vice versa] by necessity you have to make assumptions about underlying distributions, but that is not enough.

Most distributions are characterized by a combination of a measure of location [i.e. mean for normal distribution] and a measure of dispersion [i.e. standard deviation for normal distribution].

A quartile is actually a non-parametric measure of dispersion. If you have an underlying distribution that is known AND know the corresponding value for the corresponding measure of location, it is possible to calculate the corresponding quartiles

Specific answer to supplied details:

A confidence interval usually is a 95% confidence interval. That is: the chance of the 'true result' laying outside the given interval is 5%. The true ratio may be only 0,95 or as high as 1.25, but it is not likely. Its probability is less than 95%.

Now for the odds ratio's:

If you are on exactly on the 25th percentile of fat intake, the ratio of people with hypertention to those without is 1.05.

If you are on the 75th percentile, the figure is 1.24.

Are you sure this is the P-value? If the interval crosses 1.0, the P-value should be higher than 0.05.

@ Pablo Hernández-Alonso  The P-value is only used as a measure of [given a lot of assumptions] the probability that, IF the true ratio EQUALS 1.000 [infinite sample size], that you will find a distribution that deviates as much or more from unity. It does not necessarily or by definition require the ratio to be > 1.0 or < 1.0 .

This is an often encountered mistake.

Say you find a mean of 9.66 in your sample, while the [known to be accurate] population mean is 8.0 and the population spread, expressed in standard deviation is 1.0, your sample deviates 1.66 standad deviation from the population mean. [assuming a normal distribution]

If your H1 hypothesis was: My sample deviates [in any direction!!] from the reference population, than the integral from -1.66 to +1.66 [times the SD] includes 90.106% of the population. [2-tailed]

If your H1 hypothesis was: My sample mean is higher [1-tailed] than that of the reference population, then the integral from minus infinity to + 1.66 for the normal distribution includes 95.106% of the population.

The H0 hypothesis either way is: the sampe mean is not different from the population mean.

In the two tailed case, the probability that the H0 hypothesis explains your findings is 100-90.106= 9.894%, which is by convention usually not considered significant.

If you insisted on the second hypothesis [1-tailed one], the probability that H0 is true drops to 100-95.106 = 4.894%

A similar argument can be set up for the case where your sample mean was just 6.34, with the same P-values.

In two tailed comparisons, your statement is about 'as extreme, or more extreme' results. Both a ratio significantly below 1 or above 1 can be considered 'extreme'.

Only in one-tailed comparisons your logic is true, but that requires that you not only predict that a difference exists, but also that you know the direction.

In science it is normal to test 2-tailed unless you have strong case in favour of 1-tailed testing.  2-tailed is the norm. And then your logic is flawed.

@Judocus Borm: We are not talking about means, but CI for an odds ratio.

The measure of association is an odds ratio, so one would not expect the a 95% CI of 0.97 - 1.21 to be statistically significant. I am also surprised by the p-value of 0.03.

I have come across another study where OR was 1.19, 95% CI was 1.15-1.22 and p

@Sowmya Krishnan, I can not understand a significant P-value with a 95% CI interval cutting 1.00... Are you sure this is the P-value for this particular OR and it is not related to a particular p-linear trend or the p-value of the full-adjusted model?

It is the p value of the full-adjusted model

@Petter: When statistical methods are being used based on different underlying assumptions, or when underlying assumptions are nor met, weird combinations of resultas may be encountered. Quartiles are often used in a non-parametric sense, but can usually also be computed from [parametric] distributions [if the use of that distribution is warrantned]. The P-value is the probability that your results can be obtained by chance only if the conservative hypothesis H0 is true. When no further special issues are involved the H0 hypothesis would be that the odd's are 1.0.

So you are right that normally the 95% confidence interval would not include an odds ratio of 1.0 at a P of 0.03.

However, if [hypothetically] the circumstances are not perfect. I give an hypothetical example here: In Holland in certain areas we have high concentratons of foreigners living. for example some neighbourhoods with high percentages of people from Indian ancestry. With high frequency of diabetes and morbid obesitas and high intake of fat. It is quite possible that we see for example a bimodal distribution in fat intake.

Whenever the population distribution markedly deviates from the assumed statistical distribution, we are in trouble.

As fat intake has a strong effect, the Central Limit Theory can not be applied to justify for example a normal distribution,

It is not uncommon when 2 different statistical methods are applied and the P-value is not very different from the cut-off value [here 0.05] that such seemingly inconsistent results are being found.

Say [hypothetically] that the CI was been computed using resampling methods [ie Bootstrap] but the P-value derived otherwise, such inconsistencies are not unheard of and the likelyhood of encountering [seemingly] contradictory results is rather high around the cut-off value for significance.

All statistical theory starts with a whole bundle of assumptions and usually most [in biomedicine] are either not fullfilled or at the very least not verified at all.

Therefore the concept of 'robustness' is used because for many problems multiple methods are defendable but will not allways give identical results.

See Sir Maurice Kendall and A.Stuard The advanced theory of statistics vol. 1 distribution theory and

Siegel Nonparametric statistics for the behavioral sciences

so in brief: in theory  you were completely on the mark with your comment,

in real life in biosciences and around the cut-off value for significance  counter-intuitive result combinations are not unheard of.

@Judocus Borm I agree that this "counter-intuitive result combination" is possible if the CI and p-value are derived from different tests. In these cases it is however important to explain why two different tests were used. One cannot expect a reader to assume that counter-intuitive results are derived from two different tests. 

@Petter  I agree with you. I was cautioned as using quartiles and odds-ratio's are not the most obvious combinations and various non-parametric / resampling methods can be used to obtain all sorts of confidence intervals with all sorts of quantitative data.

Whereas in physics very often one gets away with using the Central limit theory to defend the use of parametric techniques, in biomedical research malignant distributions are far more likely.  In fact the comments after the main question introduced confusion. The method description and the actual data description is not exact enough for high quality answers without some quess work or adding our own assumptions. The crux lies in the unexpected finding of "significance" and a confidence interval that includes an odds ratio of 1.0.

That in this case odds ratio's vary for different quartiles in itself is to be expected, as the most likely estimate for the effect deviated from 1.0000. That the most likely estimate deviated from 1.0 itself is certainly not evidence of significance.

In the approach that is normally used in science, we accept a 5% chance that we reject the H0 hypothesis when in fact the H0 hypothesis is true.

People often think that the estimated parameters [based on a sample] of a distribution are 'the truth' whereas in fact these are themselves estimates and as such, have an error distribution of their own.

For many of these estimates there exist various different approaches or techniques or refinements.

I.e. nonparametric tests based on ranking use the underlying assumption that tied observations [with equal ranking] do not occur.

Within certain limits  various tricks can be applied that will save your ass most of the times, but these should not be confused with clean mathematical / statistical theory. and especially in case of results near the cut-off value for significance such situations can also lead to occasional counter intuitive results.

So occasional counter-intuitive results can be the result of combining different tests, or as discussed here of different corrections especially in combination with 'not so clean' or ill-distributed data. 

For example the mean and the standard deviation are not equally robust in case of a simple error such as missing a decimal point in a single result even in a large sample.

Say you have 1000 results with a mean of 1 and an s.d. of 0.1 and for a single datapoint the decimal point is forgotten. So a value of 1.2 is instead processed as 12. your mean will rise only 0.01, which is likely to go unnoticed, but your s.d. will become much larger. So the estimate of the true mean is robust for this type of data-errors. Standard deviations are not!

Such seemingly trivial things can and do cause all sorts of neat algorithms occasionally behave in an unexpected manner. If instead a resampling technique was used to obtain CI, the effect of 1 in a 1000 such errors would be much less. So in this case resampling techniques are more robust [for this specific situation. This is what robustness in statistical sense is about.

I am not a statistician, but I have been exposed to various hard core algorithms more than most during the past 30 years and I have seen quite a lot of ill behaved biomedical data.

The book by Siegel I referred is a classical one. Easy to follow and comprehend, hence its recommendation for use in the behavioral sciences and it explains  the concept of robustness quite good.