We have studied the nonlinear elasticity effects in III-N compounds. Particularly, we have determined the pressure dependences of elastic constants in wurtzite and zinc-blende InN, GaN, and AlN by performing ab-initio calculations in the framework of plane-wave pseudopotential implementation of the density-functional theory. We have found that C11, C12 in zinc-blende phase and C11, C12, C13, C33 in wurtzite phase depend significantly and almost linearly on hydrostatic pressure, for all considered nitrides. Much weaker dependences on pressure have been observed for C44 in both wurtzite and zinc-blende phases. Further, we have examined the influence of pressure dependence of elastic constant on the pressure coefficient of light emission, dEE / dP, in wurtzite and cubic, InGaN/GaN and GaN/AlGaN quantum wells. We show that the pressure dependence of elastic constants results in significant reduction of dEE / dP in nitride quantum wells and essentially improves the agreement between experimental and theoretical values.