IS there any quantum analogy where a three state (or three body) system shows chaotic dynamics as three body problem in classical mechanics?
Quantum chaos is in some sense generic behavior. This article gives two real-world examples from atomic and solid-state physics that are essentially equivalent to the hydrogen atom but with some modification made to the potential. E.g., one of them is that you get chaotic behavior in high-$n$ states in the hydrogen atom, in a strong magnetic field. These two examples can both be considered to be two-body systems. I assume you're asking about three-body systems because for Newtonian gravity, the two-body system doesn't exhibit chaos. I don't know for sure, but I assume that, e.g., a nearly-ionized hekium atom in a strong magnetic field would exhibit the same chaotic behavior as in the case of hydrogen.