van der Waals heterostructures (vdWHs) based on 2D layered materials with select properties are paving the way to integration at the atomic scale, and may give rise to new heterostructures exhibiting absolutely novel physics and versatility. Herein, we investigate the structural and contact types in a 2D vdW heterobilayer between MoGe₂N₄ and MoSi₂N₄ monolayers, and the monolayers in the presence of electrical graphene (GR) contact. In the ground state, the MoGe₂N₄/MoSi₂N₄ heterobilayer forms type-II band alignment, which effectively promotes the separation of electrons and holes and provides opportunity for further electrons and holes. Thus, the MoGe₂N₄/MoSi₂N₄ heterobilayer is promising for designing optoelectronic devices with significantly suppressed carrier-recombination. Interestingly, the insertion of the GR contact to a MoGe₂N₄/MoSi₂N₄ heterobilayer gives rise to the formation of a metal/semiconductor contact. Depending on the GR position relative to the MoGe₂N₄/MoSi₂N₄ heterobilayer, the GR-based heterostructure can form either an n-type or p-type Schottky contact. Intriguingly, the contact barriers in the GR contacted MoGe₂N₄/MoSi₂N₄ heterobilayer are significantly smaller than those in the GR contacted with MoGe₂N₄ or MoSi₂N₄ monolayers, suggesting that the GR/MoGe₂N₄/MoSi₂N₄ heterostructure offers an effective pathway to reduce the Schottky barrier, which is highly beneficial for improving the charge injection efficiency of the contact heterostructures. More interestingly, by controlling the interlayer coupling through stacking, both the Schottky barriers and contact types in the GR/MoGe₂N₄/MoSi₂N₄ heterostructure can be manipulated. Our findings could provide theoretical insight into the design of nanodevices based on a GR and MoGe₂N₄/MoSi₂N₄ heterobilayer.