Federated learning (FL) has attracted tremendous attentions in recent years due to its privacy preserving measures and great potentials in some distributed but privacy-sensitive applications like finance and health. However, high communication overloads for transmitting high-dimensional networks and extra security masks remains a bottleneck of FL. This paper proposes a communication-efficient FL framework with Adaptive Quantized Gradient (AQG) which adaptively adjusts the quantization level based on local gradient's update to fully utilize the heterogeneousness of local data distribution for reducing unnecessary transmissions. Besides, the client dropout issues are taken into account and the Augmented AQG is developed, which could limit the dropout noise with an appropriate amplification mechanism for transmitted gradients. Theoretical analysis and experiment results show that the proposed AQG leads to 25%-50% of additional transmission reduction as compared to existing popular methods including Quantized Gradient Descent (QGD) and Lazily Aggregated Quantized (LAQ) gradient-based method without deteriorating convergence properties. Particularly, experiments with heterogenous data distributions corroborate a more significant transmission reduction compared with independent identical data distributions. Meanwhile, the proposed AQG is robust to a client dropping rate up to 90% empirically, and the Augmented AQG manages to further improve the FL system's communication efficiency with the presence of moderate-scale client dropouts commonly seen in practical FL scenarios.