We studied the hydrogen adsorption on the surface of a covalently bonded bilayer borophene-graphene heterostructure decorated with Pt, Ni, Ag, and Cu atoms. Due to its structure, the borophene-graphene bilayer combines borophene activity with the mechanical stability of graphene. Based on the density functional theory calculations, we determined the energies and preferred adsorption sites of these metal atoms on the heterostructure's borophene surface. Since boron atoms in different positions can have different reactivities with respect to metal atoms, we considered seven possible adsorption positions. According to our calculations, all three metals adsorb in the top position above the boron atom and demonstrate catalytic activity. Among the metals considered, copper had the best characteristics. Copper-decorated heterostructure possesses a feasible near-zero overpotential for hydrogen evolution reaction. However, the borophene-graphene bilayer decorated with copper is unstable with respect to compression. Small deformations lead to irreversible structural changes in the system. Thus, compression cannot be used as an effective mechanism for additional potential reduction.