Reinforcement learning (RL) has shown great effectiveness in quadrotor control, enabling specialized policies to develop even human-champion-level performance in single-task scenarios. However, these specialized policies often struggle with novel tasks, requiring a complete retraining of the policy from scratch. To address this limitation, this paper presents a novel multi-task reinforcement learning (MTRL) framework tailored for quadrotor control, leveraging the shared physical dynamics of the platform to enhance sample efficiency and task performance. By employing a multi-critic architecture and shared task encoders, our framework facilitates knowledge transfer across tasks, enabling a single policy to execute diverse maneuvers, including high-speed stabilization, velocity tracking, and autonomous racing. Our experimental results, validated both in simulation and real-world scenarios, demonstrate that our framework outperforms baseline approaches in terms of sample efficiency and overall task performance.