The tailored saturated-unsaturated defect modes are realized in the proposed graphene-based hyperbolic metamaterials (HMs) in the mid-infrared and near-infrared bands by using the threshold light. In the case of saturated absorption, graphene is identified as a transparent and lossless medium. The effects of the incident angle, the number of periods, and the chemical potential of graphene are also taken into consideration. The computed results show that the greater saturated defect mode (SDM) can be observed at a smaller incident angle and a larger period. The unsaturated defect mode (USDM) cannot be affected by the incident angle while it can be controlled by the chemical potential. In addition, two novel threshold lights (0.2 MW/cm2 and 60 MW/cm2) producing bistable graphene absorption are analyzed by adopting nonlinear Kerr (NLK) materials. These results can be applied to nonlinear optical switches and logic devices.