First-principles calculations were performed to study the elastic stiffness constants ($C_{ij}$) and Debye temperature ($\theta_D$) of wurzite (wz) AlN and GaN binary semiconductors at high pressure. The lattice constants were calculated from the optimized structure of these materials. The band gaps were calculated at 𝛤 point using local density approximation (LDA) approach. The unit cell volume, lattice parameters, $c/a$, internal parameter (𝑢), elastic constant ($C_{ij}$), Debye temperature ($\theta_D$), Hubbard parameter (𝑈) and band gap ($E_g$) were studied under different pressures. The bulk modulus ($B_0$), reduced bulk modulus ($B'_0$) and Poisson ratio ($\vee$) were also calculated. The calculated values of these parameters are in fair agreement with the available experimental and reported values.