  The interplay between quantum fluctuation and spacetime curvature is shown to
induce an additional quantum-curvature (QC) term in the energy-momentum tensor
of fluid using the generalized framework of the stochastic variational method
(SVM). The QC term is necessary to satisfy the momentum conservation but the
corresponding quantum hydrodynamics is not necessarily cast into the form of
the Schr\"{o}dinger equation, differently from the case of the Euclidean
spacetime. This seems to suggest that the existence of the Hilbert space is not
a priori requirement in the quantization of curved spacetime systems. As an
example, we apply the Friedmann-Robertson-Walker (FRW) metric and show that
this effect contributes to the cosmological acceleration although it is too
small in the present non-relativistic toy model.
