Orthogonal group of degree two and type -1

Definition
Consider the finite field $$\mathbb{F}_q$$ with $$q$$ elements, $$q$$ an odd prime power. Suppose $$\alpha$$ is an element of $$\mathbb{F}_q^\ast$$ that is not a square. The orthogonal group $$O(-1,2,q)$$ or $$\Omega(-1,2,q)$$ is defined as the orthogonal group for the symmetric bilinear form given by the matrix:

$$\begin{pmatrix} 1 & 0 \\ 0 & \alpha \\\end{pmatrix}$$

In other words, as a subgroup of the general linear group of degree two, it is defined as follows:

$$\Omega(-1,2,q) := \left \{ \begin{pmatrix} a & b \\ c & d \\\end{pmatrix} \mid a^2 + b^2\alpha = 1, c^2 + d^2\alpha = \alpha, ac + bd \alpha = 0 \right \}$$