A systematic study of Brown’s characteristic curves of the two center Lennard–Jones plus point quadrupole (2CLJQ) fluid was carried out using molecular simulation and molecular-based equation of state (EOS) modeling. The model parameters (elongation and quadrupole moment) were varied systematically covering the range relevant for real fluid models. In total, 36 model fluids were studied. The independent predictions from the EOS and the computer experiments are found to be in very good agreement. Based on these results, the influence of the quadrupole moment on the fluid behavior at extreme conditions is elucidated. The quadrupole interactions are found to have a surprisingly minor influence on the extreme state fluid behavior. In particular, for the Amagat curve, the quadrupole moment is found to have an almost negligible influence in a wide temperature range. The results also provide new insights into the applicability of the corresponding states principle, which is compared to other molecular property features. Interestingly, for a wide range of quadrupole moments, the fluid behavior at extreme conditions is conform with the corresponding states principle—opposite to the influence of other molecular features. This is attributed to the symmetry of the quadrupole interactions. Moreover, an empirical correlation for the characteristic curves was developed as a global function of the model parameters and tested on real substance models. Additionally, the applicability of Batschinski’s linearity law for the Zeno curve was assessed using the results for the 2CLJQ fluid.