We explored the pressure-induced structural phase transitions and elastic properties of AuMTe2 (M = Ga, In) using the full-potential linearized augmented plane wave method within the framework of density functional theory, applying both generalized gradient and local density approximations. Thermodynamic properties were further assessed through the quasi-harmonic model. We determined the transition pressures for the phase shift from the chalcopyrite structure to the NaCl rock-salt phase in both AuGaTe2 and AuInTe2. Additionally, we calculated and analyzed mechanical properties, such as bulk modulus, shear modulus, Young's modulus, Poisson's ratio, elastic anisotropy, ductility versus brittleness, and hardness for the polycrystalline forms of AuMTe2 (M = Ga, In). The study also examined how temperature and pressure affect the Debye temperature, heat capacities, thermal expansion, entropy, bulk modulus, Grüneisen parameter, and hardness, utilizing the quasi-harmonic Debye model.