Incoherent inelastic neutron scattering experiments were performed on Na0.7CoO2 and Na0.28CoO2.1.3H2O in order to understand how the dynamics of the hydrogen-bond network of water is modified in the triangular crystalline lattice NaxCoO2.yH2O. Using quasi-elastic neutron scattering (QENS), we were able to differentiate between two types of proton dynamics: a fast process (due to water strongly bound into the sodium cobalt oxyhydrate structure during the hydration process) and a slow process (likely attributable to a collective motion). High-resolution QENS experiments, carried out on Na0.28CoO2.1.3H2O, show that, at temperatures above 310 K, the water dynamics can be well-described by a random jump diffusion model characterized by a diffusion constant equal to 0.9 x 10(-9)m2/s, which is significantly lower than the rate of diffusion for bulk water. Furthermore, our results indicate that, at room temperature, the sodium ions have no influence on the rotational dynamics of the "fast" water molecules.