In both cases you´re injecting 500pg into your LC column, so in theory you should observe the same area for the corresponding chromatographic peak, assuming you have no leaks in your injection loop or no air bubbles in your automated injection system (syringe).
As Ana mentioned, yes, in theory. In practice, there are a few things which may affect your peak and the measured area.
If the sample is being dissolved in a strong solvent in both cases, the diluted sample may have poor peak shape and some degree of "fronting" because the strong solvent carries the sample down the column with little interaction with the stationary phase until it gets diluted. The peak may then integrate to a smaller value. Assuming reverse phase HPLC, strong solvents include, but aren't limited to, methanol, acetonitrile, DMSO, or DMF.
If the sample is dissolved in the mobile phase, the peak may be wider in the diluted sample because the front of the peak is being pushed down the column as the sample is being loaded. This may not affect the integration.
If the sample is dissolved in a solvent weaker than the mobile phase (such as water, assuming reverse phase again), the compound gets focused at the head of the column, and I would expect the elution and integration to be unchanged.
Depending on the column size and flow rates, the effects above may be minimal.
In a well-behaved LC system a standard may give you this result. This is definitely NOT the case with interferents present. ESI is very sensitive to the overall sample matrix, so for samples that have other compounds eluting close to your compound of interest you may see a huge increase in peak area for your smaller injection volume. Make sure you try this in actual samples spiked with your compound before you draw this blanket conclusion.
5 ul vs 50 ul of volume is a huge change in volume. Yes, Ana's answer is a good theoretical one, but not a practical one that we use. In fact, you probably would not get the same area at all (or as good RSD) because a change in injection volume (w/ same injection solution) often effects the retention time and peak shape (basic fundamentals of chromatography). It is for this reason that when we prepare calibration standards, we always prepare them such that the exact same injection volume is used FOR ALL of the standards and samples [ https://hplctips.blogspot.com/2014/03/external-vs-internal-standard.html ]. This is standard procedure and everything gets validated at the same volume, reducing error.
For HPLC or LC/MS applications, we do not prepare a stock solution and then use different volumes to obtain the different concentrations needed. That may work fine when preparing solutions at the bench, but it does not work when preparing samples for chromatography. Injection volumes must be optimized for each method (Column Dimensions, Flow rate and Mobile Phase composition). Once optimized, a specific volume is then used for all samples/standards for validation. Don't change the volume (as you may change the Rt, linearity or peak shape). This minimizes the number of variables as only concentration changes and not volume.
BTW: Another practical reason why we would not use different injection volumes as in your example (5 ul vs 50 ul) is that each injection system has an optimized range for accurate delivery. Outside of that range, the injection volume accuracy falls off. For example, a typical 0.1 to 100 ul injector is most accurate between 20 and 80 ul and less accurate outside of those ranges. To obtain reproducible results, you want to adjust your volumes to work within the most accurate ranges (something you should verify as part of your method development).