Considering the interaction of electrons and impurities with interface optical (IO) phonons, the ionization energy of a bound polaron in wurtzite GaN/AlN strained quantum wells (QW's) is investigated theoretically by means of a modified Lee-Low-Pines variational approach, in which the strong built-in electric field (BEF) due to the spontaneous and piezoelectric polarizations of GaN/AlN QW's is included. Our numerical calculations show that the interaction between the impurity and the IO-phonon field plays an important role in screening the Coulomb interaction. Moreover, we find that the BEF has a considerable influence on the bound polaron effect in GaN/AlN QW's with a large well width. The ionization energy of a bound polaron sensitively depends on the position of the impurity center in GaN/AlN QW's. With the increasing of the well width d (d > 2.5 nm), the donor ionization energy increases monotonically and approaches a constant for the case of the donor fixed on one interface of the QW due to the BEF. On the contrary, the ionization energy decreases slowly with the increasing of the well width (d > 0.25 nm) if the BEF is ignored.