Adlayer-Substrate interfacial interactions play a crucial role in the epitaxial growth of in-plane or vertical-stacked graphene/h-BN (G/h-BN) heterostructures on metal substrates. Herein, the interface and interactions of G/h-BN on Ni(111) and Cu(111) were comparatively studied using the DFT-D2 method of Grimme, including the critical long-range van der Waals forces. Our results showed that monolayer h-BN adsorbed on both Ni(111) and Cu(111) with N atom atop site and B atom on hollow site (NtopBhcp/fcc) was energetically more favorable than that with B atom atop site and N atom on hollow site (BtopNhcp/fcc). The NtopBhcp/fcc configurations of Monolayer h-BN on Ni(111) were metallic and strongly chemisorbed by p-3d hybridization with strong charge transfer, while the BtopNhcp/fcc configurations on Ni(111) and both NtopBhcp/fcc and BtopNhcp/fcc configurations on Cu(111) were weakly metallic and physisorbed by the weak pi-3d orbitals interaction. The graphene layers adsorbing on both h-BN/Ni(111) and h-BN/Cu(111) with one C atom on top site of B atom and the other on top of hollow site (BtopNhollow) was the most energetically favorable in total energies. The adsorption energy of the G/h-BN on Ni (111) increased as the increase of the graphene layers, while it would slightly decrease as the graphene layers increased on Cu(111). On Ni(111) surface, the interlayer interaction among G/h-BN layers were not only distinctly higher than that of the G/h-BN without metal substrates, but also increase with the increase of graphene layers. On Cu(111) surface, the interlayer interaction energies among G/h-BN layers were lower than that of corresponding free G/h-BN except for the mono- and bi-layer of graphene, and gradually decrease with the increase of graphene layers. All these findings provide insight into the controlled epitaxial growth of graphene/h-BN heterostructures on metal substrates and the design of graphene-h-BN based electronic devices.