In this work we introduce modulation asymmetry and nonlinearity asymmetry to investigate the quantum synchronization dynamics in a four-node optomechanical system with nonlinear Kerr media. Through quantum Langevin equations and a generalized synchronization measure $S_q^{\varphi }$, we demonstrate that modulation asymmetry induces robust antisynchronization with a stabilized phase difference near $\pi$. Conversely, nonlinearity asymmetry enables continuous phase modulation while preserving synchronization robustness. Notably, under conditions of high nonlinearity strength, synchronization demonstrates high sensitivity to frequency but minimal dependence on coupling configurations. In the specific case without nonlinearity, however, the synchronization behavior differs significantly between the two coupling topologies. These conclusions are expected to generalize to networks with more nodes, enabling the regulation of the synchronization phase.

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• Open quantum systems
• Optomechanics
• Synchronization
• Coupled oscillators