The recent discovery of a novel hexagonal phase of GeSe (γ-GeSe) has triggered great interest in nanoelectronics applications owing to the electrical conductivity of its bulk phase being even higher than that of graphite while its monolayer is a semiconductor. For potential applications, the construction of functional two-dimensional (2D) contacts is indispensable. Herein, via first-principles calculations, we propose the design of van der Waals heterostructures (vdWHs) of γ-GeSe contacting graphene, 2D h-BN and MoS as representatives of metallic, insulator, and semiconductor partners, respectively. Our work shows that the h-BN or graphene layer donates electrons to the γ-GeSe layer, resulting in n-doping in γ-GeSe, while the MoS layer accepts electrons from the γ-GeSe layer, leading to p-doping of the latter. The γ-GeSe/BN heterostructure has a type-I band alignment with large band offsets, indicating that BN can be used as an effective passivating layer to protect γ-GeSe from environmental disturbance while maintaining its major electronic and optical characteristics. The γ-GeSe/graphene heterostructure is prone to having a very low Schottky barrier of tens of meV, easily overcome by thermal excitation, making it tunable by strain and external electric fields. The γ-GeSe/MoS vdWH forms a Z-scheme interface, which is beneficial for carrier splitting and photon utilization. Our work indicates that γ-GeSe can be well passivated by BN, and form an intimate contact with graphene for high charge injection efficiency and with MoS for efficient carrier splitting for redox reactions.