Binary and triplet distribution functions provide essential insights into the microscopic interactions between particles in hydrogen plasma. These functions describe the probability of finding particles at specific distances and velocities from each other, offering a detailed understanding of the plasma's structure and thermodynamic properties. Hydrogen plasma models are pivotal for advancing our understanding of astrophysical phenomena, such as stellar formation and solar activity, while also playing a key role in the development of fusion energy a promising clean and sustainable energy source. Additionally, these models find applications in diverse fields, including materials processing, semiconductor manufacturing, and plasma medicine. The Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy is a fundamental tool for studying the dynamics of quantum plasmas, providing a rigorous framework for understanding the behavior of many interacting quantum particles under various conditions. By capturing the correlations and collective interactions among particles, it bridges microscopic dynamics with macroscopic observables. This paper introduces a novel approach to calculating quantum binary and triplet distribution functions for hydrogen plasma models using the BBGKY hierarchy. The findings are analyzed and compared with prior studies, shedding light on their implications for both theoretical and practical applications in plasma physics and quantum systems.