  We prove a family of 3-term relations in the Grothendieck ring of the
category of finite-dimensional modules over the affine quantum algebra of type
$G_2$ extending the celebrated $T$-system relations of type $G_2$. We show that
these relations can be used to compute classes of certain irreducible modules,
including classes of all minimal affinizations of type $G_2$. We use this
result to obtain explicit formulas for dimensions of all participating modules.
