EDIT 1. $\textbf{Remark}$. Let $C,D\in SL_2(\mathbb{Z})$ satisfying $(1)$; if we want to simplify the form of $C,D$ (at least that of $C$) to simplify the required sufficient condition concerning a, b, c, then we must go through its conjugacy class wrt. $GL_2(\mathbb{Z})$. Unfortunately, to a fixed trace, can correspond several such conjugacy classes.
For example, to $tr(C)=-1$ corresponds a unique class, that of $\begin{pmatrix}0&-1\\1&-1\end{pmatrix}$; yet, when $tr(C)=12$, there are at least $2$ conjugacy classes, that of $\begin{pmatrix}0&-1\\1&12\end{pmatrix}$ and $\begin{pmatrix}7&2\\15&5\end{pmatrix}$$\begin{pmatrix}7&2\\17&5\end{pmatrix}$.
$\textbf{NB}$ EDIT 2. It remains$\textbf{The case when $A,B$ are simultaneously triangularizable}$ and where $tr(A)=a,tr(B)=b,tr(AB)=c$ are integers. Then, necessarily, $a,b,c$ are linked by a unique relation $a^2+b^2+c^2-abc-4=0$ (according to studythe @user44191 's comment below).
The above relation can be realized by integers iff $\sqrt{(b^2-4)(c^2-4)}$ is a square. In the sequel, we suppose that $a,b,c$ satisfy the above two conditions.
CASE 1. $A$ or $B$ (for example $A$) has distinct eigenvalues. Then there are $3$ similarity classes over $\mathbb{C}$.
$[diag(*,*),\begin{pmatrix}*&1\\0&*\end{pmatrix}],[diag(*,*),\begin{pmatrix}*&0\\1&*\end{pmatrix}],[diag(*,*),\begin{pmatrix}*&0\\0&*\end{pmatrix}]$.
In a second step, one calculates the similarity classes associated to $tr(A)$ over $\mathbb{Z}$ and we seek the conditions about $a,b,c$ (for the $3$ classes) as in the detailed example (the case whenof the standard class) in the first part of the post.
CASE 2. $A$ and $B$ have double eigenvalues $\pm 1$.
2.1. $A$ or $B$ (for example $A$) is equal to $\pm I_2$; then the condition $b\in\mathbb{Z}$ suffices.
2.2. $A,B$ are simultaneously triangularizableis similar over $\mathbb{C}$ to $\pm\begin{pmatrix}1&1\\0&1\end{pmatrix},\pm\begin{pmatrix}1&u\\0&1\end{pmatrix}$ where $u\not= 0$.
Proceed over $\mathbb{Z}$ as in the first part of the post.
