  We consider canonically conjugated generalized space and linear momentum
operators $\hat{x}_q$ and $ \hat{p}_q$ in quantum mechanics, associated to a
generalized translation operator which produces infinitesimal deformed
displacements controlled by a deformation parameter $q$. A canonical
transformation $(\hat{x}, \hat{p}) \rightarrow (\hat{x}_q, \hat{p}_q)$ leads
the Hamiltonian of a position-dependent mass particle in usual space to another
Hamiltonian of a particle with constant mass in a conservative force field of
the deformed space. The equation of motion for the classical phase space $(x,
p)$ may be expressed in terms of the deformed (dual) $q$-derivative. We revisit
the problem of a $q$-deformed oscillator in both classical and quantum
formalisms. Particularly, this canonical transformation leads a particle with
position-dependent mass in a harmonic potential to a particle with constant
mass in a Morse potential. The trajectories in phase spaces $(x,p)$ and $(x_q,
p_q)$ are analyzed for different values of the deformation parameter. Lastly,
we compare the results of the problem in classical and quantum formalisms
through the principle of correspondence and the WKB approximation.
