We investigate the quantum dissipative dynamics near the stable states (attractors) of a driven Duffing oscillator. A refined perturbation theory that can treat two perturbative parameters with different orders is developed to calculate the quantum properties of the Duffing oscillator near the attractors. We obtain the perturbative analytical results, that go beyond the standard linearization approach, for the renormalized level spacings, the orbital displacements, and the effective temperature near the classical attractor. Furthermore, we demonstrate that strong damping induces additional slight renormalization of level spacings, and the Bose distribution together with dephasing. Our work provides new insights into the quantum dynamics of the driven Duffing oscillator and offers a theoretical framework that can be applied to related quantum systems near their stable states.