The dynamics of water molecules confined in approximately cylindrical silica nanopores is investigated using molecular simulation. The model systems are pores of diameter varying between 20 and 40 Å containing water at room temperature and at full hydration, prepared using grand canonical Monte Carlo simulation. Water dynamics in these systems is studied via molecular dynamics simulation. The results of the basic characterization of these systems have been reported in A. A. Milischuk and B. M. Ladanyi [J. Chem. Phys. 135, 174709 (2011)]. The main focus of the present study is the self-intermediate scattering function (ISF), F S(Q, t), of water hydrogens, the observable in quasi-elastic neutron scattering experiments. We investigate how F S(Q, t) depends on the pore diameter, the direction and magnitude of the momentum transfer Q, and the proximity of water molecules to the silica surface. We also study the contributions to F S(Q, t) from rotational and translational motions of water molecules and the extent of rotation-translation coupling present in F S(Q, t). We find that F S(Q, t) depends strongly on the pore diameter and that this dependence is due mainly to the contributions to the ISF from water translational motion and can be attributed to the decreased mobility of water molecules near the silica surface. The relaxation rate depends on the direction of Q and is faster for Q in the axial than in the radial direction. As the magnitude of Q increases, this difference diminishes but does not disappear. We find that its source is mainly the anisotropy in translational diffusion at low Q and in molecular reorientation at higher Q values.