Topological phase, a novel and fundamental role in matter, displays an extraordinary robustness to smooth changes in material parameters or disorder. A crossover between topological physics and quantum information may lead to inherent fault-tolerant quantum simulations and quantum computing. Quantum features may be preserved by being encoded among topological structures of physical evolution systems. This requires us to stimulate, manipulate and observe topological phenomena at single quantum particle level, which, however, hasnt been realized yet. Here, we address such a question whether the quantum features of single photons can be preserved in topological structures. We experimentally observe the boundary states of single photons and demonstrate the performance of topological phase on protecting the quantum features in quasi-periodic systems. Our work confirms the compatibility between macroscopic topological states and microscopic single photons on a photonic chip. We believe the emerging quantum topological photonics will add entirely new and versatile capacities into quantum technologies.