Exciton effects on the third-harmonic generation in GaAs/AlGaAs infinite and finite square quantum wells are calculated within the framework of the fractional-dimensional space approach. The wave functions and bound energies are obtained as functions of spatial dimensionality and the dimension is a function of the well width. For an infinite confining potential the dimension (D) has a transition from the 3D limit to 2D limit when the well width decreases. However, in a finite well, when the well width decreases below a given value, the dimension increases. The analytical expression of the third-harmonic generation is described using the compact density method and the iterative procedure. The numerical results show that the third-harmonic generation coefficient with considering exciton effects is 40% greater than the one without considering exciton effects and it is very sensitively dependent on the exciton confinement. In addition, the smaller the relaxation constant is, the larger the third-harmonic generation will be.