This paper examines the effect of quantized communications on the convergence behavior of the primal-dual algorithm in quadratic network utility maximization problems with linear equality constraints. In our set-up, it is assumed that the primal variables are updated by individual agents, whereas the dual variables are updated by a central entity, called system, which has access to the parameters quantifying the system-wide constraints. The notion of differential entropy power is used to establish a universal lower bound on the rate of exponential mean square convergence of the primal-dual algorithm under quantized message passing between agents and the system. The lower bound is controlled by the average aggregate data rate under the quantization, the curvature of the utility functions of agents, the number of agents and the number of constraints. An adaptive quantization scheme is proposed under which the primal-dual algorithm converges to the optimal solution despite quantized communications between agents and the system. Finally, the rate of exponential convergence of the primal-dual algorithm under the proposed quantization scheme is numerically studied.