This question is related to a simple case of a previous question:
How many solutions are there for $$ x^2+y^2=1 $$ in a finite field $F_q$?
The answer of the this question would give the order of the orthogonal group $O_2({F_q})$.
But I don't see a way how I can count the solutions.
[Partial work.] Things might boil down to count $$ t=\left\{\frac{u}{u^2+v^2}\mid u,v\in F_q,\ \ u^2+v^2\not=0\right\} $$ which is what I learned from a post here. Let $k$ be the number of solutions to $u^2+v^2=0$. Then $q^2-k$ gives the number of possible denominators.