We study the nonlinear optical absorption properties of a spherical quantum dot with Hellmann potential, focusing on the two-photon absorption (TPA) process, using GaAs/AlGaAs material as an example. { The radial Schr"{o}dinger equation is solved using the Nikiforov-Uvarov method, while the two-photon absorption coefficient is determined through second-order perturbation theory concerning the electron-photon interaction}. Our study shows that the intraband transition has a smaller energy transition than the interband transition, leading to TPA spectra for the intraband transition that is restricted within a smaller energy range and exhibits a higher peak value than those for the interband transition. The peak corresponding to $\ell=2$ consistently appears to the left of the peak corresponding to $\ell=1$ in both intraband and interband transition cases. Additionally, the dependence of the absorption peak position on the order of transition, $n$, differs between intra- and inter-band transitions. We also observe blue shift behavior in the TPA spectra as all three parameters, $r_0$, $V_{0e}$, and $\eta$, increase. Our investigation has the potential to enable the design of novel photonic devices, ultra-fast optical switches, and highly efficient solar cells through the optimization of quantum dot material properties.