In non-relativistic quantum mechanics the mass can, in principle, be considered an observable and thus described by a self-adjoint operator.
In this sense a quantum physical system may have several different values of the mass and a value is fixed as soon as one performs a measurement of the mass observable, exactly as it happens for the momentum for instance.
However, it is possible to prove that, as the physical system is invariant under Galileian group (or Galilean group as you prefer), a superselection rule arises, the well-known Bargmann mass superselection rule. It means that coherent superpositions of pure states with different values of the mass are forbidden.
Therefore the whole description of the system is always confined in a fixed eigenspace of the mass operator (in particular because all remaining observables, including the Hamiltonian one, commute with the mass operator).
In practice, the mass of the system behaves just like a non-quantum, fixed parameter. This is the reason, barring subtle technicalities (non-separability of the Hilbert space if the spectrum of the mass operator is continuous), why the mass can be considered a fixed parameter rather than a self-adjoint operator in non-relativistic quantum mechanics.
In relativistic quantum mechanics the picture is quite different. First of all, one has to distinguish between elementary systems (elementary free particles in with Wigner's defintion) and compound (interacting) systems. The formers are defined as irreducible (strongly continuous) unitary representations of Poincaré group. Each such representation is identified by a set of numbers defining the eigenvalues of some observables which attains constant values in the representation because of the irreducibility requirement. The nature of these numbers depend on the structure of the group one is considering.
Each such observable, in the irreducible Hilbert space of the system has the form $\lambda I$ where $\lambda$ is a fixed real number. Referring to the Poincaré group, the mass operator turns out to be one of these elementary observables. Therefore, in relativistic quantum mechanics, the elementary systems must have the trivial mass operator, which as before, can be considered as a fixed, non-quantum parameter.
The picture changes dramatically if one focuses on compound systems: there the mass is simply the energy operator evaluated in the rest frame of the system. It generally shows a mixed spectrum made of a continuous part, due to the "relative" kinetic energy and, below that, a point spectrum describing the possible masses of the overall system.
ADDENDUM. As Arnold Neumaier pointed out to me, neutrinos appear to have non-fixed values of the mass (i.e. the mass operator is not trivial) in view of the presence of the weak interaction. In my view, it is disputable if they can be considered elementary particles since they include weak interaction in their description. Surely they are elementary from a purely physical viewpoint. Maybe Wigner's description is physically inappropriate.