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With quasi-elastic neutron scattering, we study the single-particle dynamics of the water confined in a hydrophilic silica material, MCM-41, at 4 kbar. A dynamic crossover phenomenon is observed at 219 K. We compare this dynamic crossover with the one observed at ambient pressure and find that (a) above the crossover temperature, the temperature dependence of the characteristic relaxation time at ambient pressure exhibits a more evident super-Arrhenius behavior than that at 4 kbar. Especially, at temperatures below about 230 K, the relaxation time at 4 kbar is even smaller than that at ambient pressure. This feature is different from many other liquids. (b) Below the crossover temperature, the Arrhenius behavior found at ambient pressure has a larger activation energy compared to the one found at 4 kbar. We ascribe the former to the difference between the local structure of the low-density liquid (LDL) phase and that of the high-density liquid (HDL) phase, and the latter to the difference between the strength of the hydrogen bond of the LDL and that of the HDL. Therefore, we conclude that the phenomena observed in this paper are consistent with the LDL-to-HDL liquidliquid transition hypothesis. [ABSTRACT FROM AUTHOR]