The relationship of human brain structure to cognitive function is complex, and how this relationship differs between childhood and adulthood is poorly understood. One strong hypothesis suggests the cognitive function of Fluid Intelligence (Gf) is dependent on prefrontal cortex and parietal cortex. In this work, we developed a novel graph convolutional neural networks (gCNNs) for the analysis of localized anatomic shape and prediction of Gf. Morphologic information of the cortical ribbons and subcortical structures was extracted from T1-weighted MRIs within two independent cohorts, the Adolescent Brain Cognitive Development Study (ABCD; age: 9.93 ± 0.62 years) of children and the Human Connectome Project (HCP; age: 28.81 ± 3.70 years). Prediction combining cortical and subcortical surfaces together yielded the highest accuracy of Gf for both ABCD (R = 0.314) and HCP datasets (R = 0.454), outperforming the state-of-the-art prediction of Gf from any other brain measures in the literature. Across both datasets, the morphology of the amygdala, hippocampus, and nucleus accumbens, along with temporal, parietal and cingulate cortex consistently drove the prediction of Gf, suggesting a significant reframing of the relationship between brain morphology and Gf to include systems involved with reward/aversion processing, judgment and decision-making, motivation, and emotion.