Wireless traffic prediction plays an indispensable role in cellular networks to achieve proactive adaptation for communication systems. Along this line, Federated Learning (FL)-based wireless traffic prediction at the edge attracts enormous attention because of the exemption from raw data transmission and enhanced privacy protection. However FL-based wireless traffic prediction methods still rely on heavy data transmissions between local clients and the server for local model updates. Besides, how to model the spatial dependencies of local clients under the framework of FL remains uncertain. To tackle this, we propose an innovative FL algorithm that employs gradient compression and correlation-driven techniques, effectively minimizing data transmission load while preserving prediction accuracy. Our approach begins with the introduction of gradient sparsification in wireless traffic prediction, allowing for significant data compression during model training. We then implement error feedback and gradient tracking methods to mitigate any performance degradation resulting from this compression. Moreover, we develop three tailored model aggregation strategies anchored in gradient correlation, enabling the capture of spatial dependencies across diverse clients. Experiments have been done with two real-world datasets and the results demonstrate that by capturing the spatio-temporal characteristics and correlation among local clients, the proposed algorithm outperforms the state-of-the-art algorithms and can increase the communication efficiency by up to two orders of magnitude without losing prediction accuracy. Code is available at https://github.com/chuanting/FedGCC.