In the Quadratic Reciprocity Law
$$\exists x\in\Bbb{N}\quad x^2\equiv p\pmod q\iff\exists y\in\Bbb{N}\quad y^2\equiv q\pmod p$$ if $p\equiv q\equiv 1\pmod4$.


Is there any relation between $x$ and $y$ in general?



On the quadratic reciprocity law?
It is not clear what relation you look for.


E.g., $x$ and $y$ may be viewed as reductions of the same residue $z$ modulo $q$ and $p$, respectively, where $z^2\equiv p+q\pmod{pq}$.


