In Quantum Mechanics, position is an observable, but time may be not. I think that time is simply a classical parameter associated with the act of measurement, but is there an observable of time? And if the observable will exist, what is an operator of time?
The problem of extending Hamiltonian mechanics to include a time operator, and to interpret a time-energy uncertainty relation, first posited (without clear formal discussion) in the early days of quantum mechanics, has a large associated literature; the survey article
carefully reviews the literature up to the year 2000. (The book in which Busch's survey appears discusses related topics.) There is no natural operator solution in a Hilbert space setting, as Pauli showed in 1958,
by a simple argument that a self-adjoint time operator densely defined in a Hilbert space cannot satisfy a CCR with the Hamiltonian, as the CCR would imply that $H$ has as spectrum the whole real line, which is unphysical.
Time measurements do not need a time operator, but are captured well by a positive operator-valued measure (POVM) for the time observable modeling properties of the measuring clock.
In QM, the temporal variable $t$ is not an observable in the technical sense (i.e., in the same sense that position and momentum are). In order to be an observable, it should have to exist a linear self-adjoint operator $\hat T$ whose eigenvalues $t$ were the outcomes of measurements. But then (at least in the most naive way and according with the Schr. equation) the Hamiltonian and the temporal operator should be non-compatible observables with canonical commutation relations like position and momentum. And this is not possible because in a quantum theory the Hamiltonian must be bounded from bellow and this would imply that its conjugate (time operator) were no self-adjoint.
However, there exist mean lifetimes which are quantum-mechanically computable (they are inverses of probabilities per unit of time) and have units of time. In some sense (arguably vague sense), this is a quantum notion of time.