Underground hydrogen (H₂) storage in saline aquifers is a viable solution for large-scale H₂ storage. Due to its remarkably low viscosity and density, the flow of H₂ within saline aquifers exhibits strong instability, which needs to be thoroughly investigated to ensure safe operations at the storage site. For the first time, we develop a lattice Boltzmann model tailored for pore-scale simulations of the H₂-brine system under typical subsurface storage conditions. The model captures the significant contrast of fluid properties between H₂ and brine, and it offers the flexibility to adjust the contact angle to suit varying wetting conditions. We show that the snap-off is enhanced in a system with a high capillary number and a small contact angle. These conditions lead to a low recovery factor, which is unfavorable for H₂ production from the aquifer. Moreover, the relative permeability curves, computed from the simulation results, exhibit distinct behaviors for H₂ and brine. In the case of the wetting phase, the relative permeability can be quantified using the quadratic expression, whereas for the non-wetting phase, the relative permeability exhibits a nearly linear behavior, and saturation alone appears insufficient to characterize the relative permeability at large saturations of non-wetting phase. This implies that different formula for liquid and gas phases may be employed for continuum-scale simulations.