  As a particular one parameter deformation of the quantum determinant, we
introduce a quantum $\alpha$-determinant and study the
$\mathcal{U}_q(\mathfrak{gl}_n)$-cyclic module generated by it: We show that
the multiplicity of each irreducible representation in this cyclic module is
determined by a certain polynomial called the $q$-content discriminant. A part
of the present result is a quantum counterpart for the result of Matsumoto and
Wakayama (2005), however, a new distinguished feature arises in our situation.
