You can use quantile regression to test an interaction, just like OLS regression. It's a different question, though, whether you should. What is your reason to want to use quantile regression? Do you have an a priori reason to care more about the median than the mean (e.g., are you worried your mean is being affected by outliers)? Or do you just want to try something new because the first attempt didn't get you a significant result and you're willing to keep trying different things until something works? In the former case, trying another method like quantile regression is fine; in the latter, it is basically p-hacking (see e.g. http://pss.sagepub.com/content/early/2011/10/17/0956797611417632) and is very inadvisable (although it can still be ok to do as long as you are very and honest about it when you present your results). Sometimes, unfortunately, effects we expected just don't come out, and there's nothing we can do about it.

As for centering in particular: while I agree with Christer that centering is not strictly necessary, there are times it can be helpful. This is particularly the case if you are also interested in looking at the main effects of 'b' and 'c' in addition to the interaction. Imagine that 'b' is a two-level categorical factor and 'c' is a continuous variable. The coefficient you get for 'b' will indicate the difference between two levels of 'b' at the Intercept (i.e., when c==0). This can be very misleading, since when the lines have different slopes, the difference between levels of 'b' at the intercept may be very different than the mean difference between levels of 'b'. But once you center the variables, then the difference at the intercept is the mean difference because the intercept (0) is right in the middle of the data. The attached plot illustrates this with a regression of Temperature on Wind and Season (summer vs. not-summer); on the left you have non-centered data, and the coefficient for Season looks misleadingly small because teh lines are close together near the intercept (even though the intercept is far away from most of the data). On the right you have centered data and the coefficient for Season accurately reflects the mean difference between the two groups. (Note that this issue still applies even if both variables are continuous, it's just harder to illustrate.)

You don't have to center any of your variables at all before creating interaction terms, in fact this makes them less translucent when it comes to interpretation (although their standard errors may be lower). There is principally no difference between linear and quantile/median regression in this respect :-)

You can use quantile regression to test an interaction, just like OLS regression. It's a different question, though, whether you should. What is your reason to want to use quantile regression? Do you have an a priori reason to care more about the median than the mean (e.g., are you worried your mean is being affected by outliers)? Or do you just want to try something new because the first attempt didn't get you a significant result and you're willing to keep trying different things until something works? In the former case, trying another method like quantile regression is fine; in the latter, it is basically p-hacking (see e.g. http://pss.sagepub.com/content/early/2011/10/17/0956797611417632) and is very inadvisable (although it can still be ok to do as long as you are very and honest about it when you present your results). Sometimes, unfortunately, effects we expected just don't come out, and there's nothing we can do about it.

As for centering in particular: while I agree with Christer that centering is not strictly necessary, there are times it can be helpful. This is particularly the case if you are also interested in looking at the main effects of 'b' and 'c' in addition to the interaction. Imagine that 'b' is a two-level categorical factor and 'c' is a continuous variable. The coefficient you get for 'b' will indicate the difference between two levels of 'b' at the Intercept (i.e., when c==0). This can be very misleading, since when the lines have different slopes, the difference between levels of 'b' at the intercept may be very different than the mean difference between levels of 'b'. But once you center the variables, then the difference at the intercept is the mean difference because the intercept (0) is right in the middle of the data. The attached plot illustrates this with a regression of Temperature on Wind and Season (summer vs. not-summer); on the left you have non-centered data, and the coefficient for Season looks misleadingly small because teh lines are close together near the intercept (even though the intercept is far away from most of the data). On the right you have centered data and the coefficient for Season accurately reflects the mean difference between the two groups. (Note that this issue still applies even if both variables are continuous, it's just harder to illustrate.)

Hi!

Investigating the data you have using a quantile regression method may be viable. As you do know, linear regression only fits data with a seemingly linear relation, and as for quantile regression, a different model is fit for each pth quantile, thus may account for non-linear relations between your variables. You may or may not opt to centre your variables. And what's even more fun with using quantile regression is it is robust and fairly resistant to outliers.