No. An action of a group $G$ on a graph $\Gamma$ induces an action of $G$ on $\pi_1(\Gamma)$ by outer automorphisms. So a representation $G\to \mathrm{GL}_n({\mathbb Z})$ can come from an action on a graph, only if it lifts to a homomorphism $G\to \mathrm{Out}(F_n)$ over the quotient homomorphism $\mathrm{Out}(F_n)\to \mathrm{GL}_n({\mathbb Z})$. This paper of Zimmermann gives an example of a finite cyclic subgroup of $\mathrm{GL}_n({\mathbb Z})$ that does not lift.

No. An action of a group $G$ on a graph $\Gamma$ induces a homomorphism $G\to \mathrm{Out}(\pi_1(\Gamma))=\mathrm{Out}(F_n)$. So a representation $G\to \mathrm{GL}_n({\mathbb Z})$ can come from an action on a graph only if it lifts to a homomorphism $G\to \mathrm{Out}(F_n)$ over the quotient homomorphism $\mathrm{Out}(F_n)\to \mathrm{GL}_n({\mathbb Z})$. This paper of Zimmermann gives an example of a finite cyclic subgroup of $\mathrm{GL}_n({\mathbb Z})$ that does not lift.