The geometric quadratic (\( GQ \)) index is a recently introduced degree-based topological descriptor, and Kumar et al. observed that it is potentially a very good molecular descriptor. In this paper, we characterize the extremal graphs (chemical) and trees concerning the geometric quadratic index of a given order and size. Then, we determine the \( n \)-vertex trees, unicyclic and bicyclic graphs with the maximum, the second, the third, the fourth, the fifth, and the sixth maximum geometric quadratic indices.