The energies and binding energies of the bound polarons in a wurtzite In0.19Ga0.81N/GaN quantum well are investigated by means of a modified Lee-Low-Pines variational method. Contributions of ground state binding energies and different branches of a longwave optical phonon mode to the energies and binding energies of the bound polarons as a function of the well width and impurity center position are given. Effects of the anisotropy of phonon frequency and built-in electric field in the system on the energies and binding energies, and the electron and impurity center-optical phonon interaction, are included in the calculations. Results show that the contributions of optical phonons and built-in electric field to the ground state energy and binding energy of the bound polarons in a wurtzite In0.19Ga0.81N/GaN quantum well are very large, and result in the reduction of energy and binding energy. The binding energy decreases monotonically with increasing well width, and the speed of decrease is fast in the narrower well while the speed of decrease is slow in the wider well. Contributions of different branches of phonons to the energies and binding energies as a function of well width are different. In the narrower well, contributions of the confined phonon (withoud built-in electric field) are smaller than those of the interface and half-space phonons, while in the wider well, contributions of the confined phonons are larger than those of the interface and half-space phonons. Contributions of the confined phonon (with built-in electric field) become larger, whereas those of the interface and half-space phonons become smaller, and the total contribution of phonons also have obvious change. Contributions of these optical phonons to the ground state energies and binding energies of the bound polarons in In0.19Ga0.81N/GaN quantum wells are larger than the corresponding values (about 3.11.6 meV and 1.50.3 meV) of those in GaAs/Al0.19Ga0.81As quantum wells. The binding energies in In0.19Ga0.81N/GaN quantum wells decrease monotonically with increasing location Z0 of the impurity center for a constant well width d =8 nm, and the decrease of speed becomes faster. As the position of the impurity center is increasing, the contributions of the the interface and half-space phonons decrease slowly, and those of the confined phonons increase slowly as well.