The local ordering and features of the molecular mobility of water confined in voids of a pure silica mordenite were studied using the molecular dynamics simulation over a temperature range from 298 to 163 K. The simulated system was a fragment of mordenite consisted of 2 × 2 × 4 unit cells filled with 384 water molecules. Three different water models: SPCE, SPCF, and TIP5P were considered. To study the effect of nanoconfinment, the results were compared with bulk water. The modeling suggests that at room temperature, a 2D (in c and b directions of the mordenite cell) water diffusion takes place, while upon cooling, the diffusion in b direction essentially slows down. The analysis of microstructure shows that the pores prevent the formation of a full tetrahedral structure of water environment that results in formation of several water substructures. A detailed analysis of water reorientational motion was carried out and the activation energies were determined from temperature dependences of the correlation times. Of the three water models considered, SPCE demonstrated the best performance. The results obtained can be helpful for interpretation of experimental temperature dependences of NMR relaxation rates for water molecules confined in porous media with complex topology.