Unveiling the nonlinear interactions between terahertz (THz) electromagnetic waves and free carriers in two-dimensional materials is crucial for the development of high-field and high-frequency electronic devices. Herein, we investigate THz nonlinear transport dynamics in a monolayer graphene/MoS2 heterostructure using time-resolved THz spectroscopy with intense THz pulses as the probe. Following ultrafast photoexcitation, the interfacial charge transfer establishes a nonequilibrium carrier redistribution, leaving free holes in the graphene and trapping electrons in the MoS2. When probed with intense THz pulses exceeding 34 kV/cm in a peak electric field, significant THz saturable absorption is observed over a period of 20 ps. Furthermore, the photoinduced change in the transmitted THz waveform, linked to the THz-driven nonlinear current, manifests as a substantial self-phase modulation. These nonlinear responses can be attributed to the competition between rapid carrier heating and slow carrier cooling via electron-electron and electron-phonon scattering in the charge-transfer-induced hole system of the graphene layer. This work demonstrates an integration of advantages arising from robust nonlinear absorption in graphene and enhanced photocarrier harvesting in transition metal dichalcogenides by exploiting heterostructure construction.