Summary: As a particular one parameter deformation of the quantum determinant, we introduce a quantum \(\alpha\)-determinant \(\det_q^{(\alpha)}\) 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 the authors [Alpha-determinant cyclic modules of \(\mathfrak{gl}_n\), J. Lie Theory 16, 393–405 (2006; Zbl 1102.17004)], however, a new distinguished feature arises in our situation. Specifically, we determine the degeneration of the multiplicities for ‘classical’ singular points and give a general conjecture for singular points involving semi-classical and quantum singularities. Moreover, we introduce a quantum \(\alpha\)-permanent \(\text{per}_q^{(\alpha)}\) and establish another conjecture which describes a ‘reciprocity’ between the multiplicities of the irreducible summands of the cyclic modules generated respectively by \(\det_q^{(\alpha)}\) and \(\text{per}_q^{(\alpha)}\).