The optical Kerr effect (OKE) spectroscopy measured with heterodyne detection (HD) is a useful tool to provide information regarding intermolecular vibrations and structural relaxations in liquid water. Recently, the measurements of the OKE spectroscopy have been extended to the supercooled regime of water. Though the measured results can be well described by using a phenomenological model, the time-resolved OKE spectroscopy of liquid and supercooled water still need a comprehensive understanding. In this paper, we investigated the OKE nuclear response functions of this peculiar liquid and their reduced spectral densities by performing molecular dynamics simulations with the TIP4P/2005 water model. The collective polarizability of water was computed via a dipolar induction scheme, which involves the intrinsic polarizability and the first-order hyperpolarizability tensor of water molecule. Our simulation results were qualitatively consistent with the HD-OKE experimental observations for displaying that the polarizability anisotropy relaxation of supercooled water in the high-density liquid phase was fragile-like by following a stretching exponential decay with an exponent βs insensitive to temperature and the temperature dependence of the relaxation time exhibited a power-law divergence at a singular temperature Ts with a critical exponent γs. Indicated by our quantitative results, Ts was predominately determined by the structural arrest, but βs and γs were not only related to the structural relaxation but also influenced by the collective polarizability of the liquid. For all investigations, the effects due to the first-order hyperpolarizability tensor were examined.