From the first-principles calculations with the Local Density Approximation (LDA) and Generalized Gradient Approximation (GGA) approaches, the structural, elastic, electronic and thermodynamic properties of the hexagonal superconductor Palladium monotelluride (PdTe) under high pressure have been studied. The calculated results show that the compressibility along the a-axis is lower than that along the c-axis, and the elastic constants C ij , elastic moduli (shear modulus G, bulk modulus B) and the Debye temperature Θ D of the hexagonal PdTe monotonically increase with pressure. The calculations of the ductility factors (Poisson ratio ν, G/B) and anisotropy factors (A U , A G ) show that the pressure can strengthen the ductility, and make the hexagonal PdTe transform to be anisotropic. According to the criteria of elastic stability, we can predict that the hexagonal PdTe will not be mechanically stable any more beyond the pressure of 18.9GPa. From the calculated electronic properties, we find that the pressure also makes the itinerant character of electrons stronger and the total density of states at the Fermi Level decrease. Moreover, the superconductivity is also discussed from the calculated Θ D and electronic properties. Besides, based on the quasi-harmonic Debye model, the Grüneisen parameter γ, heat capacity C V and coefficient of thermal expansion α with temperature under different pressures have been investigated in detail.