We investigate the exact quantum motions of two ions in a two-dimensional Paul trap. When the Coulomb-correlation of the system is considered, we obtain a set of exact solutions of the system and the corresponding perturbed solutions, containing the exact discrete eigenstates and eigenenergies, and the approximate piecewise continuous spectra and the band structure. Taking several quantum states with lower quantum numbers as examples, we compute the average distances between two ions and give the plots of probability distribution of the two ions.