Autonomous docking between Unmanned Aerial Vehicles (UAVs) and ground robots is essential for heterogeneous systems, yet most existing approaches target wheeled platforms whose limited mobility constrains exploration in complex terrains. Quadruped robots offer superior adaptability but undergo frequent posture variations, making it difficult to provide a stable landing surface for UAVs. To address these challenges, we propose an autonomous UAV-quadruped docking framework for GPS-denied environments. On the quadruped side, a Hybrid Internal Model with Horizontal Alignment (HIM-HA), learned via deep reinforcement learning, actively stabilizes the torso to provide a level platform. On the UAV side, a three-phase strategy is adopted, consisting of long-range acquisition with a median-filtered YOLOv8 detector, close-range tracking with a constraint-aware controller that integrates a Nonsingular Fast Terminal Sliding Mode Controller (NFTSMC) and a logarithmic Barrier Function (BF) to guarantee finite-time error convergence under field-of-view (FOV) constraints, and terminal descent guided by a Safety Period (SP) mechanism that jointly verifies tracking accuracy and platform stability. The proposed framework is validated in both simulation and real-world scenarios, successfully achieving docking on outdoor staircases higher than 17 cm and rough slopes steeper than 30 degrees. Supplementary materials and videos are available at: https://uav-quadruped-docking.github.io.