This study investigates the production of radiation-induced secondary neutrons resulting from the irradiation of a water phantom by a medical proton accelerator. Measurements were conducted using a 2-inch by 2-inch stilbene scintillation detector. To distinguish between neutrons and the gamma background, we applied a pulse shape discrimination technique utilizing the charge integration ratio (Qratio) method. Our analysis carries out for three proton beam energies: 76 MeV, 80 MeV, and 132 MeV. The findings demonstrate the feasibility of using a stilbene scintillator detector in conjunction with the Qratio method for the measurement and analysis of secondary neutrons, which are detectable in significant quantities.