This paper proposes a hybrid quantum-classical algorithm to solve a fundamental power system problem called unit commitment (UC). The UC problem is decomposed into a quadratic subproblem, a quadratic unconstrained binary optimization (QUBO) subproblem, and an unconstrained quadratic subproblem. A classical optimization solver solves the first and third subproblems, while the QUBO subproblem is solved by a quantum algorithm called quantum approximate optimization algorithm (QAOA). The three subproblems are then coordinated iteratively using a three-block alternating direction method of multipliers algorithm. Using Qiskit on the IBM Q system as the simulation environment, simulation results demonstrate the validity of the proposed algorithm to solve the UC problem.