A Markov process is a random process where the new state only depends on the old state, not anything else. This can be stretched to include almost anything, since you can expand the definition of the state to record history or whatever you want, although you may make the process much more difficult to work with mathematically. In other words, the fact that something may be a Markov process is generally not interesting, because it is a very broad definition, and doesn't guarantee any interesting properties without further assumptions.
The wavefunction in QM doesn't evolve randomly, so I would say it is not technically a Markov process. On the other hand you can create a theory of observables derived from QM that is "random" (depending on your interpretation of quantum mechanics).
You will have a hard time constructing Markov Chain that correctly models the real-time evolution of physical quantum-mechanical observables. The problem is that the transition matrix that governs quantum-mechanical evolution is exp[i t H], which is not a well-formed probability distribution.
The wavefunction in QM doesn't evolve randomly, so I would say it is not technically a Markov process. On the other hand you can create a theory of observables derived from QM that is "random" (depending on your interpretation of quantum mechanics).