$p^2 - 2 q^2 = 5039$ for primes $p, q$

Not a solution, too long for comment:

I am pretty close to finish checking the first billion prime values of $q$ with no solution so far. Currenttly I'm at $q=19,047,324,319$

To make things even worse, it seems that the solutions of the equation $p^2-2q^2=5039$ for prime $q$ are extremely rare, even if you allow $p$ to be composite. So far I have found only two such solutions:

$$p=209, \ q=139$$

$$p=6889, \ q=4871$$

Both 209 and 6889 are composite numbers (the latter also being a perfect square) so both have to be discarded. I'll let the code run over the weekend but this looks more and more like mission impossible.