The pure harmonic description of a nanomechanical resonator is limited by the resonator intrinsic nonlinear behavior that originates from its stretching effect that appears at its end rods when those are fixed. While its impact depends on the geometry of the resonator, various techniques are proposed to additionally tune and manipulate the nonlinear behavior. Resonators like nanotubes, nanowires or nanobeams allow for stronger nonlinearities and lower nonlinear onsets due to their small cross-section dimensions in comparison with their length [1]. For low temperatures of quantum optomechanics, when the resonator is investigated in the vicinity of its fundamental mode, the nonlinear proprieties are enhanced when an electrostatic gradient force is applied [2]. Remarkable experimental results in the manipulation of the near-ground state of a quantum mechanical nanoresonator, reported the observation of single-phonon states [3]. Values below unity of the second-order correlation function indicating the realization of single-phonon states have been reported for microfabricated silicon nanobeams [4]. Moreover, the nonlinear dynamics of the mechanical resonator may lead to non-classical single-phonon states, for a resonator directly coupled to an electromagnetic field [5].  Here, we investigate the quantum dynamics of a nonlinear mechanical resonator with an embedded quantum dot. The two-level quantum dot is driven by an intense laser and interacts with the mechanical vibrations of the single-mode resonator. A Duffing-like oscillator with a Kerr-type nonlinearity is considered for the mechanical resonator. The model is treated as an open-system where the resonator‘s phonon field interacts with an environmental thermal reservoir. For this model one obtains single-phonon Fock states when stronger Kerr nonlinearities are considered. The quanta distribution of the phonon field is presented in figure 1, as a function of the laser-qubit detuning. One predicts a high probability to find the mechanical resonator within a single-phonon state and low chances to find other higher Fock states.