Semiconductor saturable absorber mirrors (SESAMs) with N-ion irradiated In_0.3Ga_0.7As quantum wells are investigated by both degenerate and non-degenerate phonon spectroscopy. We found that the N-ion irradiation on quantum wells (QWs) has impact not only on the recovery time of the SESAM structures, but also on the laser-induced coherent acoustic phonons. The recovery time is shortened when more quantum wells are irradiated by N ions, down to 1.6 ps (for all-QW-irradiated sample) from 19.9 ps (for sample without N-ion irradiation). The reflectivity change induced by coherent acoustic phonons is enhanced in the sample with partially irradiated QWs, but further enhancement does not occur in the all-QW-irradiated sample. The modification of carrier concentration induced by N-ion implantation potentially accounts for the improvement of phonon-induced reflectivity change, while further enhancement is hindered by the suppression of the deformation potential effect due to the fact that the recovery time is shorter than the acoustic wave period. In the multiple quantum well region, overall consistent acoustic results are obtained in the degenerate and non-degenerate configuration, with slight differences caused by different pump photon energies or excitation functions. In the distributed Bragg reflector region, up to seven orders of the center-modes of folded acoustic phonons starting from ~31 GHz are observed in the non-degenerate configuration. The detected first-order component is evidently enhanced and detected higher- order modes are overall slightly suppressed in the all-QW-irradiated sample compared to those in the partial-QW-irradiated sample. Our results imply that N- ion irradiation distribution can be used to tailor coherent acoustic phonons, thus this can be directly applied to monitor the optimization of SESAM characteristics for mode-locking lasers.