  We derive a $q$-analogue of the matrix sixth Painlev\'e system via a
connection-preserving deformation of a certain Fuchsian linear $q$-difference
system. In specifying the linear $q$-difference system, we utilize the
correspondence between linear differential systems and linear $q$-difference
systems from the viewpoint of the spectral type. The system of non-linear
$q$-difference equations thus obtained can also be regarded as a non-abelian
analogue of Jimbo-Sakai's $q$-$P_{\mathrm{VI}}$.
