In this article, we present a novel approach to analyse the structure of complex networks represented by a quantum graph. A quantum graph is a metric graph with a differential operator (including the edge-based Laplacian) acting on functions defined on the edges of the graph. Every edge of the graph has a length interval assigned to it. The structural information contents are measured using graph entropy which has been proved useful to analyse and compare the structure of complex networks. Our definition of graph entropy is based on local edge functionals. These edge functionals are obtained by a diffusion process defined using the edge-based Laplacian of the graph using the quantum graph representation. We first present the general framework to define graph entropy using heat diffusion process and discuss some of its properties for different types of network models. Second, we propose a novel signature to gauge the structural complexity of the network and apply the proposed method to different datasets.