The finite subgroups of $GL_3(\mathbb{Z})=PGL_3(\mathbb{Z})$ are known in the literature:
$\qquad$ Tahara: On the finite subgroups of $GL(3,\mathbb{Z})$. Nagoya Math. J. 41(1971), 169-209.
In particular Proposition 3 states that there are exactly two non-conjugate subgroups of order three. Representants are $$\langle \begin{pmatrix}1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & -1 \end{pmatrix}\rangle, \qquad \begin{pmatrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix}\rangle$$