Autonomous Vehicles (AVs) use multiple sensors to gather information about their surroundings. By sharing sensor data between Connected Autonomous Vehicles (CAVs), the safety and reliability of these vehicles can be improved through a concept known as cooperative perception. However, recent approaches in cooperative perception only share single sensor information such as cameras or LiDAR. In this research, we explore the fusion of multiple sensor data sources and present a framework, called CoBEVFusion, that fuses LiDAR and camera data to create a Bird's-Eye View (BEV) representation. The CAVs process the multi-modal data locally and utilize a Dual Window-based Cross-Attention (DWCA) module to fuse the LiDAR and camera features into a unified BEV representation. The fused BEV feature maps are shared among the CAVs, and a 3D Convolutional Neural Network is applied to aggregate the features from the CAVs. Our CoBEVFusion framework was evaluated on the cooperative perception dataset OPV2V for two perception tasks: BEV semantic segmentation and 3D object detection. The results show that our DWCA LiDAR-camera fusion model outperforms perception models with single-modal data and state-of-the-art BEV fusion models. Our overall cooperative perception architecture, CoBEVFusion, also achieves comparable performance with other cooperative perception models.