Metal hydrides (MHs) are promising materials for long-term hydrogen storage due to their moderate operating pressures and temperatures. Conventional large-scale hydrogen storage typically involves either liquefying hydrogen at extremely low temperatures (around -253 °C) or compressing it to high pressures (up to 700 bar). Both methods require strict safety measures to prevent leakage, otherwise it can result in significant losses of hydrogen during transport - reportedly as high as 10%. In contrast, MH-based storage systems incorporate hydrogen atoms into the metal crystal lattice in a reversible process that effectively eliminates leakage under normal operating conditions. This improves safety and storage efficiency. In addition to stationary storage, MHs also have the potential to be used in automotive fuel cell applications. In this work, a coupled multiphysics computational model is developed to capture the hydrogen absorption and desorption behavior in a representative metal hydride system. The governing equations are formulated within a porous media framework to describe the interaction between gas transport, heat transfer, and mechanical deformation of the solid skeleton. Spatial discretization is performed using the standard Bubnov-Galerkin formulation within the Finite Element Method (FEM). Owing to the presence of multiple coupled physical variables such as displacement, temperature, pore gas pressure, and mixture density, the multipatch Isogeometric Analysis (IGA) approach is employed for both accurate geometric representation and field interpolation, as well as to facilitate refinement via knot insertion and order elevation. Thermal contact effects between the hydride bed and the storage tank wall are incorporated to account for interfacial heat exchange. Selected numerical examples, validated through comparison with experimental data, demonstrate the accuracy and predictive capability of the proposed computational model.