In this paper, individual split Au square rings were numerically proposed as novel substrates for surface enhanced Raman and hyper-Raman scattering (SERS and SEHRS) simultaneously. The peak wavelengths of their localized surface plasmon resonances (LSPR) are revealed to fall in the near-infrared and visible light region, respectively, which are able to be finely tuned to match well with the wavelengths of the incident laser and hyper Raman scattered light beams. Their SEHRS and SERS performances along with electromagnetic (EM) field distributions are numerically investigated by finite element method. With the enhancement of near electric-fields generated by LSPRs, the maximum SEHRS and SERS enhancement factors are demonstrated to reach 1.22×1012 and 108, respectively. Meanwhile, the corresponding SERS based refractive-index (RI) sensitivity factor reaches as high as 258nm/RIU and 893nm/RIU, at visible and near-infrared wavelengths, respectively. The proposed structure is believed to hold great promise both for developing SEHRS, SERS and SERS based RI sensing substrates, which shows strong potential applications in nano sensing and enhanced Raman scattering.