Deep understanding of $\gamma$-ray production from the fast neutron reaction in water is crucial for various physics studies at large-scale water Cherenkov detectors. We performed test experiments using quasi-mono energetic neutron beams ($E_n = 30$ and 250~MeV) at Osaka University's Research Center for Nuclear Physics to measure $\gamma$-rays originating from the neutron-oxygen reaction with a high-purity germanium detector. Multiple $\gamma$-ray peaks which are expected to be from excited nuclei after the neutron-oxygen reaction were successfully observed. We measured the neutron beam flux by using a liquid scintillator for the cross section measurement. With a spectral fitting analysis based on the tailored $\gamma$-ray signal and background templates, we measured cross sections for each observed $\gamma$-ray component. The results will be useful to validate neutron models employed in the on-going and future water Cherenkov experiments.