Answer:
Suppose to the contrary that there are positive integers p and q that satisfy the given equation.
Note that 4p^2 - q^2 = (2p - q)(2p + q). The factors of 25 are 1, 5, 25, so we can check possible cases.
If 2p - q = 2p + q = 5, then this results in p not being an integer or q = 0, depending on how you solve the system of equations. In either case, we have a contradiction.
If 2p - q = 1 and 2p + q = 25, then p turns out to be a fraction. Similar conclusion occurs when you switch the order.
By exhaustion, we conclude that there are no such positive integers p and q.