We have theoretically studied exciton states and photoluminescence spectra of the strained wurtzite Al_xGa_1-xN/GaN quantum-well heterostructures. The electron energy spectrum is found through numerical solving of the one-band Schrödinger equation, while the hole and exciton energy spectra are determined from the Schrödinger equation with the non-symmetrical 6-band Hamiltonian, including deformational and spin-orbit interaction. The oscillator forces are calculated and the exciton states are determined, which are active in the process of the optical absorption. Three quantitative conclusions are made in a good agreement with experiments for different values of x as concerns: (i) the red shift of the zero-phonon photoluminescence (PL) peaks with increase of the quantum-well width, (ii) the relative intensities of the zero-phonon and one-phonon PL peaks, found within the non-adiabatic approach, and (iii) the values of the PL decay time as a function of the quantum-well width.