The Number of involutory matrices over $\mathbb{Z_p} $

Question

I want to prove the number of 2-by-2 Involutory Matrices ($A^2=I$) over $\mathbb{Z_p}$ using quadratic residue and legendre symbol.
I already know that the formula is $p^2$ for characteristic of a field is 2 and $p^2+p+2$ for $p$ being an odd prime as stated here: Involutory matrix $2 \times 2$
I have also shown that the number of Involutory Matrices is twice the number of solutions ($a,b$) to the congruence : $a^2+bc \equiv 1$ (mod $p$)
Is there any way to come up with the formula using quadratic residue and legendre symbol? Thanks in advance.