This study examines a single wurtzite ZnO/ZnBeO quantum well structure oriented in the polar c-direction. The focus is on investigating the binding energies associated with an impurity donor atom within this system. To achieve this, a self-consistent solution to the Schrödinger and Poisson equations is obtained using the finite difference method. The framework employed involves the effective mass and envelope function approximations. The impurity is represented using a hydrogenic-type wave function, and donor binding energies are determined via a variational approach. The research analyzes the binding energies of the 1s and 2p± states, along with the transition energy between them. These quantities are explored as functions of the well width and Be concentration, considering donor positions at the right and left interfaces as well as at the well's center. Furthermore, the impact of an external magnetic field oriented along the growth direction is assessed, spanning up to 10T, in order to quantify changes in the binding energies. The presence of a built-in electric field induces an asymmetric band profile and a triangular well configuration. This asymmetry results in a loss of symmetry within the binding energy curves. Ultimately, the investigation culminates in the computation of the oscillator strength governing transitions between the donor states. When the donor is situated at the right interface, the energy values remain relatively constant as the well width increases, and the oscillator strength values demonstrate a consistent linear rise.