Fine-tuning pre-trained foundation models has made significant progress in music information retrieval. However, applying these models to beat tracking tasks remains unexplored as the limited annotated data renders conventional fine-tuning methods ineffective. To address this challenge, we propose HingeNet, a novel and general parameter-efficient fine-tuning method specifically designed for beat tracking tasks. HingeNet is a lightweight and separable network, visually resembling a hinge, designed to tightly interface with pre-trained foundation models by using their intermediate feature representations as input. This unique architecture grants HingeNet broad generalizability, enabling effective integration with various pre-trained foundation models. Furthermore, considering the significance of harmonics in beat tracking, we introduce harmonic-aware mechanism during the fine-tuning process to better capture and emphasize the harmonic structures in musical signals. Experiments on benchmark datasets demonstrate that HingeNet achieves state-of-the-art performance in beat and downbeat tracking