Collective modes propagating in a moving superfluid are known to satisfy wave equations in a curved space-time, with a metric determined by the underlying superflow. We use the Keldysh technique in a curved space-time to develop a quantum geometric theory of fluctuations in superfluid hydrodynamics. This theory relies on a "quantized" generalization of the : two-fluid description of Landau and Khalatnikov, where the superfluid component is viewed as a quasi-classical field coupled to a normal component - the collective modes/phonons representing a quantum bath. This relates the problem in the hydrodynamic limit to the "quantum friction" problem of Caldeira-Leggett type, By integrating out the phonons, we derive stochastic Langevin equations describing a coupling between the superfluid component and phonons. These equations have the form of Euler equations with additional source terms expressed through a fluctuating stress-energy tensor of phonons. Conceptually, this result is similar to stochastic Einstein equations that arise in the theory of stochastic gravity. We formulate the fluctuation-dissipation theorem in this geometric language and discuss possible physical consequences of this theory. (C) 2018 Elsevier Inc. All rights reserved.