Difference between a quadratic residue and a number is a quadratic residue

Question

Let $p$ be an odd prime, and let $a\in\mathbb{Z}_p^*$, i.e. $a\not\equiv 0$. Let $q_1,\ldots,q_{\frac{p-1}{2}}$ be the quadratic residues mod $p$. Is there a way of knowing how many $j\leq \frac{p-1}{2}$ satisfy $q_j - a$ is a quadratic residue?