My setting is as follows: Fix a Markoff form $f_m(x,y)$ (see definition in the link below). If $f_m$ has the form ${\alpha}x^2+{\beta}xy+{\gamma}y^2$ then each element $A\in SL_2(Z)$ acts on $f_m$ in the following way: $Af_m=f_m(ax+by, cx+dy)$ ($a,b,c,d$ are the elements of the matrix $A$). Denote $G:=\{A\in SL_2(Z) : Af_m=f_m\}/\{I,-I\}$.
Question: Why is $G$ an infinite cyclic group?
For the definition of a Markoff form, see the first page here: http://arxiv.org/pdf/1106.1844v1.pdf
Edit: In order to avoid conufsion, my original question was about general quadratic forms. As the general question seems to have a negative answer (see rschwieb's answer below), I've changed it to the case of Markoff forms.