If you’re only looking at non-relativistic quantum mechanics, no, it does not. The proper way to add special relativity to quantum mechanics leads to Quantum Field Theory, and that is the theory that properly takes into account all the effects of relativity. There were attempts to modify the Hamiltonian of the non-relativistic theory to make it relativistic, but it turns out that all such theories don’t work. To add relativity you need to change the dynamical objects of the theory from particles, all the way to fields, and this is a much larger change than just adding relativity to classical mechanics.
If you’re only looking at non-relativistic quantum mechanics, no, it does not. The proper way to add special relativity to quantum mechanics leads to Quantum Field Theory, and that is the theory that properly takes into account all the effects of relativity. There were attempts to modify the Hamiltonian of the non-relativistic theory to make it relativistic, but it turns out that all such theories don’t work. To add relativity you need to change the dynamical objects of the theory from particles, all the way to fields, and this is a much larger change than just adding relativity to classical mechanics.