To the extent that you think of Brownian motion as a random walk, the natural quantum extension is the quantum random walk.
For a physics perspective, see Quantum random walks - an introductory overview, but you might prefer the more math-oriented exposition of Martin boundary theory of some quantum random walks and On algebraic and quantum random walks.
We give a concise prescription of the concept of a quantum random walk (QRW), using the example of QRW on integers as paradigm. It briefly explains the notion of quantum coin system and the coin tossing map, and summarizes two emblematic properties of that walk, namely the quadratic enhancement of its diffusion rate due to quantum entanglement between the walker and the entropy increase without majorization effect of its probability distributions. We conclude with a group theoretical scheme of classification of various known QRW's.
I understand the question is more oriented towards the physical phenomenon of Brownian motion, which I would describe as the effect of an environment with a large (infinite) number of degrees of freedom on the dynamics of a particle with a few degrees of freedom. We would then be seeking a quantum theory of friction, diffusion, and thermalization. The seminal paper here is the path integral theory of Caldeira and Leggett. The literature is very extensive, an older but still relevant review is Quantum Brownian Motion: The Functional Integral Approach.
The quantum mechanical dynamics of a particle coupled to a heat bath is treated by functional integral methods and a generalization of the Feynman-Vernon influence functional is derived. The extended theory describes the time evolution of nonfactorizing initial states and of equilibrium correlation functions. The theory is illuminated through exactly solvable models.