Distributed algorithms are the key to enabling effective large scale distributed control systems, which present many challenges due to lack of access to global information. This paper studies the effects of quantized communication among agents for distributed averaging algorithms. The agents rely on peer-to-peer interactions to exchange and update their local agreement variables that converge to the average of initial values asymptotically. Our previous work had shown that, when the underlying communication graph is static with certain types of uniform quantization effects, an optimized distributed selection of algorithm parameters results in the convergence of the agreement variables to a small neighborhood around the actual average, independent of the network size. In this paper, we extend this result to time-varying (dynamic) graphs. We first show, using an example, that the deviation from the desired average caused by quantized communication on time-varying graphs can be arbitrarily large even when the graph is connected at every iteration. This is contrary to the case without quantized communication. We then present a large class of randomized time-varying communication graphs for which the convergence to a small neighborhood around the average is guaranteed in the presence of quantized communication.