  A high order linear $q$-difference equation with polynomial coefficients
having $q$-Hahn multiple orthogonal polynomials as eigenfunctions is given. The
order of the equation is related to the number of orthogonality conditions that
these polynomials satisfy. Some limiting situations when $q\to1$ are studied.
Indeed, the difference equation for Hahn multiple orthogonal polynomials given
in \cite{Lee} is corrected and obtained as a limiting case.
