We present a novel terahertz (THz) quantum cascade laser (QCL) design where Γ-valley states are used for lasing transition and X-valley states—in particular, Xz-states—are used as injector subbands. Since the lasing states in our proposed structure are populated and depopulated mainly through the interface roughness assisted Γ-Xz electron scattering, we present a model to describe this intervalley carrier transport. In the injector region of the proposed THz QCL, we use a quaternary AlGaAsP material to introduce tensile strain, which plays a crucial role in increasing the gain. To compensate the strain per period, we propose to grow the periodic heterostructure on a GaAs0.94P0.06 virtual substrate. To simulate the carrier transport and hence calculate the gain and lasing performance of the proposed THz QCL, we use a simplified density matrix formalism that considers resonant tunneling, dephasing, and the important intersubband scattering mechanisms. Since electron temperature significantly varies from lattice temperature for QCLs, we take their difference into account using the kinetic energy balance method. We show that the proposed structure is capable of lasing up to a maximum lattice temperature of ∼119 K at 4.8 THz. For future improvements of the device, we identify major performance-degrading factors of the proposed design.