  The Racah problem for the quantum superalgebra $\mathfrak{osp}_{q}(1|2)$ is
considered. The intermediate Casimir operators are shown to realize a
$q$-deformation of the Bannai-Ito algebra. The Racah coefficients of
$\mathfrak{osp}_q(1|2)$ are calculated explicitly in terms of basic orthogonal
polynomials that $q$-generalize the Bannai-Ito polynomials. The relation
between these $q$-deformed Bannai-Ito polynomials and the
$q$-Racah/Askey-Wilson polynomials is discussed.
