  The quantum $\alpha$-determinant is defined as a parametric deformation of
the quantum determinant. We investigate the cyclic
$\mathcal{U}_q(\mathfrak{sl}_2)$-submodules of the quantum matrix algebra
$\mathcal{A}_q(\mathrm{Mat}_2)$ generated by the powers of the quantum
$\alpha$-determinant. For such a cyclic module, there exists a collection of
polynomials which describe the irreducible decomposition of it in the following
manner: (i) each polynomial corresponds to a certain irreducible
$\mathcal{U}_q(\mathfrak{sl}_2)$-module, (ii) the cyclic module contains an
irreducible submodule if the parameter is a root of the corresponding
polynomial. These polynomials are given as a $q$-deformation of the
hypergeometric polynomials. This is a quantum analogue of the result obtained
in our previous work [K. Kimoto, S. Matsumoto and M. Wakayama,
Alpha-determinant cyclic modules and Jacobi polynomials, to appear in Trans.
Amer. Math. Soc.].
