It is easy to verify that (5/p) = (p/5).
We know that (p/5) = 1 when p is quadratic residue modulo 5.
So p = 1(mod 5) or 4 (mod 5). Therefore, for every prime p in
the arithmetic progression 1+5j, 5 is residue. Similarly,
for every prime p in the progression 4+5j, 5 is residue.
We know from Diritchlet theorem that there are infinitely many primes
any arithmetic progression a+bj, for fixed co-prime pair (a,b).
So just search for first few primes in the progressions 1+5j, 4+5j.
(you can see that first few primes with this property
are 11, 41, 29, ...etc).
To my knowledge, there is no algorithm that performs better than brute force technique for finding primes in the given arithmetic progression.