  An algebraic interpretation of the $q$-Meixner polynomials is obtained. It is
based on representations of $\mathcal{U}_q(\mathfrak{su}(1,1))$ on
$q$-oscillator states with the polynomials appearing as matrix elements of
unitary $q$-pseudorotation operators. These operators are built from
$q$-exponentials of the $\mathcal{U}_q(\mathfrak{su}(1,1))$ generators. The
orthogonality, recurrence relation, difference equation, and other properties
of the $q$-Mexiner polynomials are systematically obtained in the proposed
framework.
