Ice is the most fundamental hydrogen-bonded system in which the hydrogen nuclear quantum effect significantly impacts the structure and relevant thermochemical and spectroscopic properties. While ice was experimentally investigated using proton nuclear magnetic resonance spectroscopy more than 40 years ago, the corresponding theoretical investigations have been rarely reported due to the difficulty in evaluating how the proton nuclear quantum effect influences the spectral characteristics of such a condensed material. In this study, we applied a combination of the ONIOM and multicomponent molecular orbital (MC_MO) methods for calculating the anisotropic and isotropic components of the nuclear magnetic shielding tensor of the hexagonal ice crystal to quantify the effects of nuclear quantum fluctuations on the spectroscopic properties of ice. The nuclear magnetic shielding values computed by incorporating the hydrogen nuclear quantum effect reasonably agree with the experimental values. The nuclear quantum effects were found to increase the anisotropic component of the magnetic shielding tensor while decreasing the isotropic component. Such a difference can be explained by their distinct dependence on the electrostatic field and hydrogen-bonding structural parameters.