Integrating non-terrestrial networks (NTNs) with terrestrial networks (TNs) is key to enhancing coverage, capacity, and reliability in future wireless communications. However, the multi-tier, heterogeneous architecture of these integrated TN-NTNs introduces complex challenges in spectrum sharing and interference management. Conventional optimization approaches struggle to handle the high-dimensional decision space and dynamic nature of these networks. This paper proposes a novel hierarchical deep reinforcement learning (HDRL) framework to address these challenges and enable intelligent spectrum sharing. The proposed framework leverages the inherent hierarchy of the network, with separate policies for each tier, to learn and optimize spectrum allocation decisions at different timescales and levels of abstraction. By decomposing the complex spectrum sharing problem into manageable sub-tasks and allowing for efficient coordination among the tiers, the HDRL approach offers a scalable and adaptive solution for spectrum management in future TN-NTNs. Simulation results demonstrate the superior performance of the proposed framework compared to traditional approaches, highlighting its potential to enhance spectral efficiency and network capacity in dynamic, multi-tier environments.