<p>This paper theoretically investigates the steady-state quantum correlations in a hybrid system comprising an optical cavity, a mechanical resonator, and a quantum well exciton. Our analysis is based on the covariance matrix formalism derived from linearized quantum Langevin equations, allowing us to quantify both entanglement and Gaussian quantum steering. We show that while strong bipartite entanglement is generated in both red- and blue-detuned regimes, genuine tripartite entanglement is optimally produced under red-detuning and is strongly suppressed otherwise. This reveals the crucial role of the red-detuned interaction in distributing quantum correlations across the entire hybrid network. Furthermore, we analyze quantum steering and uncover a strong asymmetry dictated by the dissipation hierarchy of the subsystems. We show that the high-Q mechanical resonator can effectively steer both the cavity and the exciton in a one-way fashion, while the reverse steering is suppressed.</p>