Training generative adversarial networks (GAN) in a distributed fashion is a promising technology since it is contributed to training GAN on a massive of data efficiently in real-world applications. However, GAN is known to be difficult to train by SGD-type methods (may fail to converge) and the distributed SGD-type methods may also suffer from massive amount of communication cost. In this paper, we propose a {distributed GANs training algorithm with quantized gradient, dubbed DQGAN,} which is the first distributed training method with quantized gradient for GANs. The new method trains GANs based on a specific single machine algorithm called Optimistic Mirror Descent (OMD) algorithm, and is applicable to any gradient compression method that satisfies a general $δ$-approximate compressor. The error-feedback operation we designed is used to compensate for the bias caused by the compression, and moreover, ensure the convergence of the new method. Theoretically, we establish the non-asymptotic convergence of {DQGAN} algorithm to first-order stationary point, which shows that the proposed algorithm can achieve a linear speedup in the parameter server model. Empirically, our experiments show that our {DQGAN} algorithm can reduce the communication cost and save the training time with slight performance degradation on both synthetic and real datasets.