In the last forty years, many scientists used graph theory to develop mathematical models for analyzing structures and properties of various chemical compounds. In this paper, we will establish formulas and bounds for generalized first Zagreb Index and coindex, which are based on degrees of vertices. In addition, for triangle and quadrangle free graphs, we will establish formulas and bounds for generalized first leap Zagreb Index and coindex, which are based on 2-distance degrees of vertices. Additionally, we will establish sharp bounds of generalized first Zagreb index and the leap index for various types of graphs and provide examples for which the sharp bounds are attained. In addition, we will find regression models and compare the first Zagreb index and the first leap Zagreb index for predicting some physicochemical properties of certain chemical compounds, benzenoid hydrocarbons.