  We solve the connection problem of a certain system of linear $q$-difference
equations recently introduced by K. Park. The result contains the connection
formulas of the $q$-Lauricella hypergeometric function $\varphi_{D}$ and those
of the $q$-generalized hypergeometric function ${}_{N+1}\varphi_{N}$ as special
cases. Our result gives a simultaneous extension of them.
