To address the degradation of trajectory tracking accuracy and the lag in obstacle avoidance for unmanned aerial vehicles (UAVs) operating in complex dynamic environments—caused by model mismatch and uncertainty in obstacle motion—a cooperative tracking and avoidance control method based on the fusion of dual Gaussian processes (GP) and model predictive control (MPC) is proposed. First, a hierarchical dynamics model of the quadrotor UAV is constructed, wherein a Gaussian process is employed to learn unmodeled dynamics online and compensate for associated errors, thereby improving prediction model accuracy and quantifying model uncertainty. Second, a Gaussian process-based trajectory prediction model for dynamic obstacles is established, in which the mean and covariance of obstacle position predictions are transformed into time-varying three-dimensional ellipsoidal probabilistic safety constraints; these constraints are subsequently embedded into the receding-horizon optimization of MPC through a unified linearization method, enabling proactive avoidance of dynamic obstacles. Furthermore, a dual GP-MPC controller that integrates dynamics compensation with obstacle prediction information is designed to collaboratively accomplish trajectory tracking and dynamic obstacle avoidance within a unified optimization framework. Simulation results demonstrate that the proposed method effectively enhances trajectory tracking accuracy, strengthens dynamic obstacle avoidance capability, and preserves favorable trajectory continuity and control smoothness in complex uncertain environments.