Classical motion of a single damped ion confined in a Paul trap is usually described by a damped harmonic oscillator model. We report the treatment of quantum damping motion of the system via a non-Hermitian Hamiltonian with dipole and quadrupole imaginary potential. By deriving and analyzing the exact solution of the system, we obtain the different real energy spectra and stable quantum states for the PT symmetry and asymmetry cases, as well as the imaginary spectrum and decaying quantum state for the PT asymmetry case. The corresponding imaginary energy parameter region and the survival probability are investigated. We find that the non-Hermitian system parameters of the external filed uniquely determine the quantum stable states and lead to the new characteristic of the morphology of wave function. Based on these properties, we propose a method of incoherently manipulating quantum transitions between the quantum stable states. By setting the decayed expectation value of ion position to be the same as the decayed displacement of the classical damped harmonic oscillator, we obtain the correspondence between the imaginary potential strength and the classical damping parameters. The results will enrich the quantum dynamics of the damped trapped ions, which may be useful in a wide application field.