Phonons, the quantum mechanical description of lattice vibrations, play a critical role in determining the thermal, electrical, and superconducting properties of quantum materials. Understanding phonon behavior is essential for advancing technologies related to energy conversion, quantum computing, and spintronic device. This dissertation explores electron-phonon coupling (EPC), anharmonic phonon-phonon scattering, phonon-magnon coupling, and nonequilibrium phenomena through advanced theoretical frameworks, computational methods, and experimental techniques, focusing on materials exhibiting significant quantum phenomena. We begin with a detailed introduction to phonon basics. Then, we introduce the many-body formalism of phonons, including theoretical concepts such as Green’s functions, Boltzmann transport equations, anharmonic phonon-phonon and electron-phonon scattering theory, and Eliashberg theory of superconductivity. Following this theoretical foundation, the dissertation presents original research across multiple studies: First, we investigate anisotropic EPC in tungsten ditelluride (WTe2), elucidating temperature-induced Lifshitz transitions via phonon linewidth analysis and EPC matrix elements. These findings have significant implications for thermal transport and electronic topology in quantum materials. Next, we present a combined computational and experimental study of thermal transport in cubic germanium telluride (GeTe), addressing the puzzling experimental observation of increased lattice thermal conductivity with rising temperature. Our analysis reveals second-nearest neighbor bond strengthening, coherent (tunneling) phonon transport effects, and critical anharmonic phonon-phonon scattering processes, contributing novel insights into thermal transport near phase transitions.Lastly, an ab initio approach is developed to investigate nonequilibrium quasiparticle-phonon dynamics in superconductors, specifically targeting Josephson junction-based transmon qubits. We demonstrate how slight deviations from equilibrium phonon distributions significantly amplify quasiparticle populations, predominantly driven by longitudinal acoustic phonons. These insights are crucial for mitigating quasiparticle-induced decoherence in quantum computing applications. This dissertation highlights the pivotal role phonons play in quantum materials and modern condensed matter physics, effectively bridging theoretical insights with experimental observations. By exploring EPC, anharmonic phonon-phonon scattering, and phonon-quasiparticle dynamics through advanced theoretical frameworks, computational techniques, and cutting-edge experiments, we deepen the understanding of fundamental quantum phenomena. The comprehensive investigations presented here not only elucidate critical physical mechanisms but also establish a solid foundation for future explorations and technological innovations in condensed matter physics, spintronics, and materials science.