We have applied first-principle calculations, based on the density functional theory, to analyze the stability, electronic and magnetic properties of monolayers of graphene with a nanodomain of boron nitride (h-BN) with different geometries. To this end, we have investigated the effects of the adsorption of one and two hydrogen atoms by different atoms at the edge of the h-BN nanodomain, where we used the GGA approximation for the functional exchange and correlation. Specifically, we found that the structures with triangular-shaped nanodomains are the most stable and have a non-integer magnetic moment upon adsorption of the hydrogen atoms. We have also shown that both the electronic and magnetic properties can be influenced by the type of atom which adsorbs the hydrogen at the edge of the nanodomain. Additionally, we have demonstrated that some structures can quench the spin that would be generated by the adsorption of the hydrogen atoms.