Hybrid Unmanned Aerial Underwater Vehicles (HUAUVs) have emerged as platforms capable of operating in both aerial and underwater environments, enabling applications such as inspection, mapping, search, and rescue in challenging scenarios. However, the development of novel methodologies poses significant challenges due to the distinct dynamics and constraints of the air and water domains. In this work, we present persistent monitoring tasks for HUAUVs by combining Deep Reinforcement Learning (DRL) and Transfer Learning to enable cross-domain adaptability. Our approach employs a shared DRL architecture trained on Lidar sensor data (on air) and Sonar data (underwater), demonstrating the feasibility of a unified policy for both environments. We further show that the methodology presents promising results, taking into account the uncertainty of the environment and the dynamics of multiple mobile targets. The proposed framework lays the groundwork for scalable autonomous persistent monitoring solutions based on DRL for hybrid aerial-underwater vehicles.