  Let $(p_n)_n$ be either the $q$-Meixner or the $q$-Laguerre polynomials. We
form a new sequence of polynomials $(q_n)_n$ by considering a linear
combination of two consecutive $p_n$: $q_n=p_n+\beta_np_{n-1}$, $\beta_n\in
\RR$. Using the concept of $\D$-operator, we generate sequences $(\beta_n)_n$
for which the polynomials $(q_n)_n$ are orthogonal with respect to a measure
and common eigenfunctions of a higher order $q$-difference operator.
