Flexible job-shop scheduling problems (FJSPs) represent one of the most complex combinatorial optimization challenges. Modern production systems and control processes demand rapid decision-making in scheduling. To address this challenge, we propose a quantum computing approach for solving FJSPs. We propose a quadratic unconstrained binary optimization (QUBO) model to minimize the makespan of FJSPs, with the scheduling scheme encoded in the ground state of the Hamiltonian operator. The model is solved using a coherent Ising machine (CIM). Numerical experiments are conducted to evaluate and validate the performance and effectiveness of the CIM. The results demonstrate that quantum computing holds significant potential for solving FJSPs more efficiently than traditional computational methods.