Summary: The non-generic central elements of the quadratic algebra \({\mathcal A}_ 1\) associated with the quantum group \(\text{GL}(2)_ q\) are found in the case where \(q\) is a root of unity. A \(q\)-boson realization of \({\mathcal A}_ 1\) is constructed. In terms of the \(q\)-boson realization the representations of \({\mathcal A}_ 1\) on the \(q\)-Fock space are studied in both generic and non-generic cases and the cyclic representation is obtained in the non-generic case.