Real-time point cloud compression is critical to time-sensitive applications of robotics such as localization, detection, and segmentation. However, previous methods usually model LiDAR point clouds (LPCs) as voxel structures and apply deep neural networks to classify the voxel occupancy and compress the voxel values, which brings unacceptable time consumption. To address this problem, we propose a hierarchical plane-quadric surface fitting-based framework for real-time LPC compression, which flexibly represents varying geometric complexities. Specifically, we first project the raw LPCs into small and compact 2D range images (RIs) by utilizing the inherent physical properties of LiDAR, and then extract the key frame of sequences. To enable adaptive modeling for different geometric regions of the intra frame, we employ the coefficient of determination as a goodness-of-fit measure, simultaneously guiding the choice between planar approximation and quadratic surface fitting. Besides, we further design a multi-frame temporal fitting strategy to exploit spatiotemporal coherence that iteratively fits the surface of point clouds with a similar range value across consecutive frames, reducing redundant computations and enhancing overall encoding efficiency. Experimental results show that our method outperforms the previous state-of-the-art works on the KITTI benchmark, achieving up to 20.61% bit-rate savings compared with PCL baselines while keeping real-time encoding performance.