= 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 $\gamma\ne0$. If $x_0\ne0$, choose any integral lift of $y_0$ and apply <Hensel's lemma> to the <polynomial> $aX^2+by_0^2-c$ in $X$: its <derivative> at the reduced <root of a polynomial>[root] is $2\alpha x_0\ne0$. If $y_0\ne0$, use the other variable instead. The nonzero <derivative> uses $p\ne2$. Thus a solution lifts to $\boxed{x,y\in\mathbb Z_p}$ as required.
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