Graphene is considered to be one of the most promising reinforcement phases for nanocomposites due to its unique two-dimensional planar structure with excellent mechanical properties. After the design of origami, the 2D material will obtain a negative Poisson’s ratio in the in-plane direction and become a metamaterial with unusual mechanical properties. Inspired by this, an origami pattern is adopted for graphene at an atomic scale using a molecular dynamics (MD) approach, and then origami graphene is embedded into a single-crystal copper matrix to obtain origami graphene/copper nanocomposites with auxetic behaviors. In the modeling, the periodic boundary condition is chosen to exhibit the Poisson’s ratio of the whole system. Under the isothermal–isobaric ensemble, the interactions between C-C, Cu-Cu, and C-Cu atoms are, respectively, determined by three potential functions: AIREBO, EAM, and LJ. The effect of the origami graphene/copper interfacial gap on the critical strain of incremental Poisson’s ratio, critical strain of engineering Poisson’s ratio, and moduli of the origami graphene/copper nanocomposites is studied to determine the optimum distance between the two phases. The influences of the mass fraction of carbon atom and temperature on those properties are discussed in detail after the MD model is confirmed.