In this paper, we propose a distributed subgradient-based method over quantized and event-triggered communication networks for constrained convex optimization. In the proposed method, each agent sends the quantized state to the neighbor agents only at its trigger times through the dynamic encoding and decoding scheme. After the quantized and event-triggered information exchanges, each agent locally updates its state by a consensus-based subgradient algorithm. We show a sufficient condition for convergence under summability conditions of a diminishing step-size.