In the dynamic fields of mathematical chemistry and chemoinformatics, caterpillar trees are widely employed as a foundational tool for modeling the structures of unbranched benzenoid hydrocarbon molecules. They are a renowned topological index used to model QSPR and QSAR between molecules. This article explores into an intriguing class of caterpillar trees that integrates randomness, offering a fresh perspective on the Zagreb index and its generalizations. We meticulously compute the skewness and kurtosis associated with the Zagreb index, presenting analytical expressions for the expected value of its generalized form. Moreover, the asymptotic behavior of the generalized Zagreb index is characterized and analyzed for caterpillars generated randomly through the application of the Martingale Central Limit Theorem. Finally, we conduct thorough numerical experiments that analyze the behaviors of statistical parameters in this unique class.