Probability that an attacker wins the discrete logarithm game when exponents are drawn from a subset
Assume there exists an adversary for your q-subset DL problem that will be successful with probability $\mathsf{succ(\lambda)}$.
If you just use this adversary to attack the original DL problem, he will get an input that comes from the set it expects with probability $q/p$ and will therefore be succesfull with probability at least $q/p \cdot \mathsf{succ(\lambda)}$.
However we know that the DL problem is hard.
Thus, we can bound the success probability of the adversary:
$$\mathsf{negl}(\lambda) \ge q/p \cdot \mathsf{succ(\lambda)}$$
$$p/q\cdot\mathsf{negl}(\lambda) \ge \mathsf{succ(\lambda)}$$

Thus we get a bound for the success probability against your q-subset DL problem.

Whether this bound is helpful, depends on $q$. As long as $p/q$ is polynomial, the upper bound is still a negligible function. So at least in theory this is still a hard problem.

In practice and for a fixed group size that is of course a completely different story.

Of course, if $p/q$ is superpolynomial, then the all bets are off and this upper bound does not help you in any way.