The following argument is inspired by tomasz's answer. Set $ B:=\begin{pmatrix}0 & -1 \\ 1 & 0\end{pmatrix},\quad G:=\text{SL}(2,\mathbb Z),\quad H:=G/\{\pm\ I\}, $ ($I$ being the identity matrix), and let $\pi:G\to H$ be the canonical projection.
Since $B$ and $-B$ are conjugate in ${GL}(2,\mathbb Z)$, it suffices to verify that $\pi(A)$ and $\pi(B)$ are conjugate in $H$.
The action of $G$ on the upper half-plane $U$ of $\mathbb C$ by Möbius transformations induces a faithful action of $H$ on $U$. It is not hard to check that an element of finite order of $H$ has a fixed point on $U$. Then, using Theorem VII.1 of Serre's A Course in Arithmetic, one sees that the order two elements of $H$ are conjugate, as required.
EDIT. The following is implicit above and in tomasz's answer:
Two elements of $\text{SL}(2,\mathbb Z)$ having the same finite order are conjugate in $\text{GL}(2,\mathbb Z)$.
Let me also point out that the proof of Theorem VII.1 in Serre's A Course in Arithmetic is short, self-contained, and (is it necessary to add?) wonderfully written.