Pell's equation and binary hyperbolic forms.

Question

We define the function $f:\mathbb Z_p \times \mathbb Z_p \rightarrow \mathbb Z_p$ such that $f(x,y)=x^2-cy^2$ and $0\neq c=a^2+b^2$.
Is it true that $f$ is hyperbolic?
In other word, is there any matrix $D$ s.t $D^t F D = H$? When
$$F= \begin{pmatrix} 1 & 0\\ 0 & -c\\ \end{pmatrix} H= \begin{pmatrix} 1 & 0\\ 0 & -1\\ \end{pmatrix}$$
I know that $f$ represent all elements of $\mathbb Z_p$.