This paper proposes a distributed algorithm over quantized communication networks for unconstrained optimization with smooth cost functions. We consider a multi-agent system whose local communication is represented by a fixed and connected graph. Each agent updates a state and an auxiliary variable for the estimates of the optimal solution and the average gradient of the entire cost function by a consensus-based optimization algorithm. The state and the auxiliary variable are sent to neighbor agents through a uniform quantizer. We show a convergence rate of the proposed algorithm with respect to the errors between the cost at the time-averaged state and the optimal cost. Numerical examples show that the estimated solution by the proposed quantized algorithm converges to the optimal solution.