Hydrogen bonds are of paramount importance in the chemistry of clays, mediating the interaction between the clay surface and water, and for some materials between separate layers. It is well-established that the accuracy of a computational model for clays depends on the level of theory at which the electronic structure is treated. However, for hydrogen-bonded systems, the motion of light H nuclei on the electronic potential energy surface is often affected by quantum delocalization. Using path integral molecular dynamics, we show that nuclear quantum effects lead to a relatively small change in the structure of clays, but one that is comparable to the variation incurred by treating the clay at different levels of electronic structure theory. Accounting for quantum effects weakens the hydrogen bonds in clays, with H-bonds between different layers of the clay affected more than those within the same layer; this is ascribed to the fact that the confinement of an H atom inside a layer is independent of its participation in hydrogen-bonding. More importantly, the weakening of hydrogen bonds by nuclear quantum effects causes changes in the vibrational spectra of these systems, significantly shifting the O-H stretching peaks and meaning that in order to fully understand these spectra by computational modeling, both electronic and nuclear quantum effects must be included. We show that after reparameterization of the popular clay forcefield CLAYFF, the O-H stretching region of their vibrational spectra better matches the experimental one, with no detriment to the model's agreement with other experimental properties.