We propose a scheme to generate quantum entanglement and Einstein–Podolsky–Rosen (EPR) steering in a hybrid qubit-cavity optomagnonic system. This hybrid system consists of a microwave cavity, a yttrium-iron-garnet (YIG) sphere, and a superconducting qubit. Due to the existence of the effective parametric (beamsplitter) coupling between the optical mode and the magnon mode (superconducting qubit), the quantum entanglement and EPR steering between the magnon and qubit modes can be achieved. It is found that bipartite entanglement and one-way quantum steering are limited to a narrow range of parameters, while two-way quantum steering appears in a wide range of parameters. Furthermore, we demonstrate that the entanglement between the magnon and the collective dressed modes, as well as the conversion of one-way quantum steering, can also be achieved. Our scheme may provide a feasible approach to exploring quantum information processing.