In this paper, we prove that if a graph does not contain any cycle of length greater than \(4\), then the square of its line graph is perfect. As an application, we give a concise proof of a known result: the strong chromatic index of a bipartite graph that does not contain any cycle of length greater than \(4\) is at most \(\Delta^2\), where \(\Delta\) represents the maximum degree of the graph. This latter result provides a partial affirmative answer to some known conjectures on upper bounds for the strong chromatic index of graphs.