The equations that describe fluid flow, the Navier Stokes equations, are generally considered chaotic. In particular turbulence is defined as " fluid motion characterized by chaotic changes in pressure and flow velocity". This means that even when the behaviour and characteristics of the phenomenon (the fluid) are entirely deterministic, the equations still give rise to chaotic behaviour.
What this means is that even when the behaviour of a system is completely deterministic, very small uncertainties in the initial conditions of a fluid can give rise to very large differences in its behaviour. With simple solid mechanics, such as the trajectory of a ball under gravity, a small uncertainty in the initial velocity of a ball gives rise to only a small uncertainty in the final trajectory of the ball. With fluids a small uncertainty in the initial velocity of a fluid, and even very small uncertainties in a calculation, can give rise to a very large difference in the answer. This is what makes turbulent flow problems hard to solve. You need extreme accuracy, extremely low errors and thus extreme computing power in order to get even reasonable accurate solutions to turbulent flow problems.
This behaviour is the case even when the behaviour of the fluid is completely deterministic. There is no need to invoke quantum or other random processes in order to see the unpredictable "chaotic" behaviour of fluids. While quantum effects do technically act on fluids, like on all material things, their effects are far too small to have a noticeable effect on fluid flow predictions, just like they are too small to affect solid mechanics. They are not considered in regular calculations.
TLDR: Fluid flow equations are deterministic (i.e. not random) but are chaotic. This is what makes them hard to solve. Quantum or other random processes have no significant effect.