Graph theory plays a significant role in understanding the structure of complex systems in many areas, including chemistry, biology, computer science, and network theory. One key part of graph theory is the study of topological indices, which are numbers that describe important features of graph structures. In quantum mechanics, chemistry, and material science, quantum graphs have emerged as a significant topic as they demonstrate quantum systems using graphs. The collaboration of quantum mechanics and graph theory has been key to introducing quantum graphs, which offer unique ways to model and understand quantum systems. The ideas of traditional graph theory have been extended to develop quantum graphs with properties like superposition and entanglement. This chapter discusses quantum graphs and two of their main properties, such as superposition and entanglement. We have introduced some topological indices for quantum graphs and proved a few theorems. At last, we have concluded the book chapter with potential future works.