Hybrid quantum systems harness the distinct advantages of different physical platforms, yet their integration is not always trivial due to potential incompatibilities in operational principles. Here, we theoretically propose and demonstrate a scheme for generating non-Gaussian mechanical states using a strongly driven hybrid system that combines cavity quantum electrodynamics (QED) and cavity optomechanics. Our protocol prepares a non-Gaussian cavity state in the dispersive regime of cavity QED and subsequently transfers it to a mechanical oscillator using the optomechanical interaction enhanced by a coherent cavity drive. While non-Gaussian cavity state control in cavity QED is well established in the dispersive regime, its behavior under strong cavity drive, essential for cavity optomechanics, remains largely unexplored. To bridge this gap, we develop an efficient simulation framework to model cavity QED dynamics in the high-photon-number regime. We show that a strong cavity drive can coherently displace the cavity state with minimal perturbations, effectively decoupling it from the qubit. The resulting large coherent cavity field enhances the optomechanical coupling strength, enabling high-fidelity transfer of non-Gaussian cavity states to the mechanical mode. These results reveal new dynamical features of driven cavity QED and open a pathway toward realizing non-Gaussian mechanical quantum memories and sensors.