Spectra of the most general Hermitian Hamiltonian which is bilinear in the creation and annihilation operators of a q-harmonic oscillator are investigated with the help of the factorization method. It is shown that there are two factorization schemes leading to discrete spectra of complicated forms. For q-oscillator models with continuous spectrum the Hamiltonian may have normalizable eigenstates of infinite multiplicity. Existence of the continuous spectrum in the interacting system is also discussed.