Nonlinear optical properties of a Zn1-xCdxO/ZnO quantum well structure originating from the transitions between the subbands within its conduction band are studied in terms of its dimension, Cd concentration and doping level. The intersubband transition energies are calculated within the effective mass and envelope function approximations. The Schrodinger and Poisson equations are self-consistently solved by considering the free and the bound charge distributions. The many-body effects, such as depolarization and excitonic shifts, are taken into account. The results are represented as functions the Cd concentration, limited between 0.05 and 0.2, and the well width, changing from 10 to 50 angstrom. The calculations indicate an intersubband transition energy range from 40 (9.67) to 138 meV (33.4 THz) and a blue-shift in it, as a result of the many-body effects, up to 11 meV in case of a doping level of 1 x 10(25) m(-3). It is shown that the transition energy increases with the widening quantum well within the range of the parameters as a result of the strong built-in electric field. The nonlinear absorption lineshape is found be to asymmetric and has the peak positions depending on the light intensity and the doping level inside the quantum well.