Focusing on the original and extended indenofluorene frameworks, we theoretically investigate the interplay between the open-shell character, the aromaticity, and the second hyperpolarizabilities gamma. Interestingly, the odd-electron density distribution, which illustrates the spatial contribution of the open-shell character, is found to well correlate with the magnetic shielding tensor distribution, which indicates the magnetic criteria of the aromaticity. This can be explained with the partial destruction of the pi-delocalization due to the emergence of odd (unpaired) electrons. Further investigation on the gamma values, which are third-order nonlinear optical (NLO) properties at the molecular scale, reveals the correlation of strong enhancement of gamma and its density distribution to the intermediate open-shell character and its odd-electron density distribution. These results will contribute not only to the detailed understanding of the structure NLO property relationships in the indenofluorene frameworks but also to building new design guidelines for highly efficient NLO molecules based on the open-shell character-aromaticity correlation.