Let $p$ be an odd prime. I am interested in how many quadratic residues $a$ sre there such that $a+1$ is also a quadratic residue modulo $p$. I am sure that this number is $$ \frac{p-6+\text{mod}(p,4)}{4}, $$ but I have neither proof nor reference. It is a particular case of the question in the title: if $a$ and $b$ are quadratic residues modulo $p$, when is $a+b$ also a quadratic residue modulo $p$?

I came into this question when counting the number of diophantine $2$-tuples modulo $p$, that is, the number of pairs $\{ a,b\}\subset \mathbb{Z}^*_p$ such that $ab+1$ is a quadratic residue modulo $p$.