If I want to determine whether $X_1$ comes from an exponential distribution and X2 comes from an exponential distribution, does it suffice to have a single Q-Q plot for the entire data set ($X_1$∪$X_2$)? Or do we need to make separate Q-Q plots for baseline and post-baseline data. I have very few observations in the baseline data so it doesn't seem to make sense to make a Q-Q plot for $X_1$.
If $X_1$ and $X_2$ potentially have different characteristics, you don't want to treat them as a single population in the plot (which might help you assess any difference).
As @whuber notes, you can do both plots on one graph.
I am using a test that depends on distributional assumptions of $X_1$ and $X_2$. Can I assume that I cannot make distributional assumptions based on the qq-plot?
You might make assumptions based on almost anything - but they're assumptions, not facts. No Q-Q plot - nor any other display or test - can tell you two things come from any given distribution, and it cannot tell you whether two samples share a population distribution.
It might tell you when such assumptions are untenable, it won't tell you that they're true. (In practice, usually pondering the exact truth of a null is a nonsensical idea. In most situations, we know before we test that the null is strictly false.)
This would imply that I would need to use a non-parametric version of the test.
Yet people make parametric distributional assumptions regularly. (While it's common to do so, the advisability of doing so in so many situations is less clear).
Generally speaking, a well chosen nonparametric test is often a better idea than a parametric one; in many cases the power gain at even a modest deviation from the parametric assumptions can be substantial.
One possibility is to do a resampling-based (permutation/randomization or possibly bootstrap) test of an otherwise parametric test. You'll usually have good properties when the assumptions hold without it being important to the type-I validity (that is it will have level-robustness when the distribution isn't of that form). It won't necessarily beat out a 'standard' nonparametric test on power if the assumptions are far enough 'out', depending on specific circumstances.