  In this paper, we construct two lattices from the $\tau$ functions of
$A_4^{(1)}$-surface $q$-Painlev\'e equations, on which quad-equations of ABS
type appear. Moreover, using the reduced hypercube structure, we obtain the Lax
pairs of the $A_4^{(1)}$-surface $q$-Painlev\'e equations.
