On quadratic residues.

Question

If $w^2 \equiv p\pmod q$ holds where $q\equiv p\equiv1\pmod 4$ primes, is there explicit reasonably succinct expressions $f,g$ such that $f(p,q,w)\equiv \pmod p$ and $g(p,q,w)\equiv q\pmod p$ holds corresponding to the two roots of $x^2\equiv p\bmod p$?