This paper focuses on the distributed optimization problem under the constraints of communication limitations and feedback delays. To address the challenges of communication constraints and feedback delays, we propose an enhanced zero-gradient-sum (ZGS) algorithm incorporating both quantized event-triggered mechanisms and feedback delay compensation.  A Lyapunov-based stability analysis is conducted through the construction of an energy function to rigorously prove the convergence of the algorithm, and comprehensive numerical simulations validate the effectiveness of the algorithm in solving distributed optimization problems under communication limitations and feedback delays.