  Minimal affinizations form a class of modules of quantum affine algebras
introduced by Chari. We introduce a system of equations satisfied by the
$q$-characters of minimal affinizations of type $G_2$ which we call the
M-system of type $G_2$. The M-system of type $G_2$ contains all minimal
affinizations of type $G_2$ and only contains minimal affinizations. The
equations in the M-system of type $G_2$ are three-term recurrence relations.
The M-system of type $G_2$ is much simpler than the extended T-system of type
$G_2$ obtained by Mukhin and the second author. We also interpret the
three-term recurrence relations in the M-system of type $G_2$ as exchange
relations in a cluster algebra constructed by Hernandez and Leclerc.
