This answer addresses the question posed in the comments. 

It follows from Bass-Serre theory that every finite subgroup of $G$ is conjugate into $SL_2(\mathbb{Z})$ (see Section II.1.4 of Serre's book [Trees][1]). So maximal finite subgroups will be order $4$ or $6$, generated by an element of trace $0$ or $1$, and thus property (M) holds. 

The second property (NM) is false in general. Consider the matrix $\left[\begin{array}{cc}a & b \\-b & a\end{array}\right]$, $a^2+b^2=1$, then it normalizes the maximal finite subgroup generated by $\left[\begin{array}{cc}0 & 1 \\-1 & 0\end{array}\right]$. Such matrices are plentiful coming from Pythagorean triples. 


  [1]: http://link.springer.com/book/10.1007/978-3-642-61856-7