The structural stability, elastic and thermodynamic properties of the intermetallic compounds MgAg, Mg4Zn8 and Ag8Mg4Zn4 in Ag-Mg-Zn alloy under high pressure and high temperature are investigated by employing the first-principles method based on the density functional theory. The theoretical results are generally in good agreement with experimental results and similar to the theoretical calculations. The calculated results indicate that the intermetallic compounds AgMg, Mg4Zn8 and Ag8Mg4Zn4 are elastically stable at zero temperature and pressure. Mg4Zn8 and Ag8Mg4Zn4 are of the ductility phase, and AgMg is of the brittleness phase. The plasticity of Ag8Mg4Zn4 is the best in the three intermetallic compounds, and AgMg is the worst. Based on the quasi-harmonic Debye model the vibrational internal energy, vibrational Helmholtz free energy, vibrational entropy, heat capacity of constant volume, heat capacity at constant pressure, thermal expansion coefficient Grneisen parameter and Debye temperature of the intermetallic compounds in Ag-Mg-Zn alloy under high pressure and high temperature are all discussed.