Are there infinitely many Pythagorean triples with these constraints?
Right triangles with one leg and the hypotenuse of prime length were investigated by Dubner and Forbes.
The prime legs are listed at https://oeis.org/A048161 with the first 10000 examples at https://oeis.org/A048161/b048161.txt
The hypotenuses are listed at https://oeis.org/A067756 with the first 10001 at https://oeis.org/A067756/b067756.txt
It is conjectured that there are infinitely many of these. However, there is still no resolution of the question, are there infinitely many primes $n^2 + 1?$ I cannot imagine that any more is known about primes $(n^2 + 1)/ 2,$ where this time $n$ would be odd; evidently considered by Euler: these $n$ are listed at https://oeis.org/A002731 . Your condition actually asks about $(p^2 + 1)/ 2 = q,$ with both $p,q$ prime. No-one knows.