  We propose a modification of a recently introduced generalized translation
operator, by including a $q$-exponential factor, which implies in the
definition of a Hermitian deformed linear momentum operator $\hat{p}_q$, and
its canonically conjugate deformed position operator $\hat{x}_q$. A canonical
transformation leads the Hamiltonian of a position-dependent mass particle to
another Hamiltonian of a particle with constant mass in a conservative force
field of a deformed phase space. The equation of motion for the classical phase
space may be expressed in terms of the generalized dual $q$-derivative. A
position-dependent mass confined in an infinite square potential well is shown
as an instance. Uncertainty and correspondence principles are analyzed.
