I want to show that $|O^{-}(2,q)|= 2(q+1)$, When $q$ is odd prime power order.
I consider $V$ be a vector space on $F=GF(q)$ and $(V,f)$ be a orthogonal space.
I want to use matrix of form $f$ but I don't know what to say about the non squre element of $F$.
A matrix $\left(\begin{array}{cc}w&x\\y&z\end{array}\right)$ lies in $O^-(2,q)$ if and only if
$$w^2-ax^2=1,\ \ \ wy-axz=0,\ \ \ y^2-az^2=-a.$$
That $a$ is a non square element of $F$.
For the first equation, there are two solutions with $(1,0)$, $(-1,0)$ with $x=0$ and, for each $w \in F$, there are two solutions $(w,x)$ with $x \ne 0$ if and only if $w^2-1$ is a non-square.
It is a standard result in Number Theory that there are exactly $(q+1)/2$ values of $w \in F$ with $w^2-1$ a square, and hence $(q-1)/2$ with $w$ a non-square.
So there are a total of $q+1$ solutions $(w,x)$ to the first equation.
You can check that, for each such $(w,x)$, $(y,z)$ satisfy the second and third equations if and only if $(y,z) = (ax,w)$ or $(-ax,-w)$, so we have a total of $2(q+1)$ such matrices.