  Recently, a birational representation of an extended affine Weyl group of
type $A_{mn-1}^{(1)}\times A_{m-1}^{(1)}\times A_{m-1}^{(1)}$ was proposed with
the aid of a cluster mutation. In this article we formulate this representation
in a flamework of a system of $q$-difference equations with $mn\times mn$
matrices. This formulation is called a Lax form and is used to derive a
generalization of the $q$-Garnier system.
