 Solution

Let $\alpha ,\beta ,\gamma$ be the nonzero reductions of $a,b,c$. The square set in ${\mathbb{F}}_p$ has $(p+1)/2$ members, including zero. Multiplication by $\alpha$, and separately the transformation $z\mapsto \gamma -\beta z$, preserve this cardinality. Hence the two sets in the hint have total cardinality $p+1>p$ and must intersect. This gives a solution $(x_0,y_0)$ modulo $p$.

The solution cannot have both coordinates zero since $\gamma \ne 0$. If $x_0\ne 0$, choose any integral lift of $y_0$ and apply Hensel's lemma to the polynomial $a{X}^2+b{y}_0^2-c$ in $X$: its derivative at the reduced root is $2\alpha x_0\ne 0$. If $y_0\ne 0$, use the other variable instead. The nonzero derivative uses $p\ne 2$. Thus a solution lifts to $\boxed{x,y\in {\mathbb{Z}}_p}$ as required.