<s>[INST] Are the number of solutions to $ax^2+bxy+cy^2\equiv u\pmod{p}$, $(x,y)\in\{0,\dotsc,p-1\}$, the same for all units $u$? Let $p$ be an odd prime and $F={\mathbb Z}/p{\mathbb Z}$.   With $a,b,c\in F$, let $Q(x,y)=ax^2+bxy+cy^2$ where $p\nmid b^2-4ac$.    I wish to prove that the number of solutions $(x,y)\in F^2$ of
$$ax^2+bxy+cy^2=u$$ is the same for all units $u\in F$.
For example, when $p=5$, the equation $x^2-2xy+3y^2=u$ has one solution when $u=0$ and 6 solutions for each of $u=1,2,3,4$.
 [/INST] Inspired by @JoeSilverman's answer, I've updated this answer to handle any odd-characteristic field $F$, and to remove a few ugly case distinctions from the original argument; but I do assume that the characteristic is not $2$, as @KConrad points out.
Put $\theta = \dfrac{b + \sqrt{b^2 - 4a c}}2$, where we have chosen a square root of $b^2 - 4a c$ that does not equal $-b$ (either one, if $4a c \ne 0$).
We have that $a x^2 + b x y + c y^2 = a x^2 + (\theta + a c\theta^{-1})x y + c y^2$ equals $(a x + \theta y)(x + c\theta^{-1}y)$.  If $\theta$ belongs to $F$, then the change of variables $\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} a & \theta \\ 1 & c\theta^{-1} \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix}$ is non-singular (with determinant $a c\theta^{-1} - \theta = (a c - \theta^2)\theta^{-1} = \sqrt{b^2 - 4a c}$), so we are considering solutions $(x', y')$ of $x'y' = u$, which form an $F^\times$-torsor.  (In particular, if $F$ is finite, then there are $\lvert F^\times\rvert$ of them.)
If, on the other hand, $\theta$ does not belong to $F$, then $b^2 - 4a c$ is not a square in $F$, so $a$ is non-$0$.  Then $a x^2 + bx y + cy^2$ equals $a^{-1}(a x + \theta y)(a x + a c\theta^{-1} y) = a^{-1}N_{F[\theta]/F}(a x + \theta y)$, so parameterising solutions of $a x^2 + b x y + c y^2 = u$ is the same as parameterising solutions of $N_{F[\theta]/F}(a x + \theta y) = u a$.  Since $(x, y) \mapsto a x + \theta y$ is an isomorphism of $F$-vector spaces (not of $F$-algebras) $F \oplus F \to F[\theta]$, we are parameterising the fibre of the norm map over $u a$.  In general, of course, fibres are either empty or torsors for $\ker N_{F[\theta]/F}$; and, if $F$ if finite, so that the norm map is surjective, then all fibres are torsors, hence have size $\lvert F[\theta]^\times\rvert/\lvert F^\times\rvert$.
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