Adopting a numerical method of solving self-consistently the Schrödinger equation and Poisson equation, the eigenstates and eigenenergies of electrons (holes) in a two-dimensional electron-hole gas are obtained for wurtzite asymmetric ZnO/MgxZn1-xO single quantum wells (QWs). In our computation, a realistic heterostructure potential is used, in which the influences from energy band bending, material doping and the built-in electric field induced by spontaneous and piezoelectric polarizations are taken into account. Furthermore, based on the Fermi's golden rule, the optical absorptions of electronic interband transitions in QWs, and their size and ternary mixed crystal effects are discussed. The results indicate that the increase of the Mg component in MgxZn1-xO enhances the build-in electric field, which forces electrons (holes) to approach to the left (right) barrier. This causes the interband transition absorption peak to decrease exponentially and to be blue-shifted. For different widths of QWs, the calculated results show that absorption peak decreases and transition energy shows a red shift with the increase of well width. The above conclusions are expected to give a theoretical guidance for improving the opto-electronic properties of materials and devices made of heterostructures with suitable optical absorption spectra and wave lengths.