Let $\alpha \in \mathbb{C}$ be an algebraic number. If the minimal plynomial of $\alpha$ over $\mathbb{Q}$ has degree $2$, we say $\alpha$ is a quadratic number. Then $\alpha$ is a root of a unique polynomial $ax^2 + bx + c \in \mathbb{Z}[x]$ such that $a > 0$ and gcd$(a, b, c) = 1$. $D = b^2 - 4ac$ is called the discriminant of $\alpha$. Since $D \equiv b^2$ (mod $4$), $D \equiv 0$ (mod $4$) or $D \equiv 1$ (mod $4$). Conversly suppose $D$ is a non-square integer such that $D \equiv 0$ (mod $4$) or $D \equiv 1$ (mod $4$). Then there exists a quadratic number $\alpha$ whose discriminant is $D$.
Let $\Gamma = SL_2(\mathbb{Z})$. Let $\sigma = \left( \begin{array}{ccc} p & q \\ r & s \end{array} \right)$ be an element of $\Gamma$. Let $z \in \mathbb{C}$. Suppose $rz + s \ne 0$. We write $$\sigma z = \frac{pz + q}{rz + s}$$ Let $D \gt 0$ be a non-square integer such that $D \equiv 0$ (mod $4$) or $D \equiv 1$ (mod $4$). We denote by $\mathfrak{RQ}(D)$ the set of real quadratic numbers of discriminant $D$. By this question, $\mathfrak{RQ}(D)$ is $\Gamma$-invariant. The following question is a real version of this question.
My question Let $\alpha, \beta$ be explicitly given elements of $\mathfrak{RQ}(D)$. Is there algorithm for solving the following problems? If yes, what is it?
Determine whether there exists $\sigma \in \Gamma$ such that $\alpha = \sigma \beta$.
If there exists such $\sigma$, determine the set $\{\sigma \in \Gamma\ |\ \alpha = \sigma \beta\}$.