We develop a framework for Laplace operators on quantum graphs. Our construction combines a distinguished inner product, with respect to which the projection onto the underlying operator system is orthogonal, with structural properties of quantum graphs under Schur multiplication and matrix transposition. This leads to a natural quantum analogue of the classical incidence operator and the associated Laplace operator. We study its basic properties, compare it with a recently proposed notion of the quantum Laplacian, and illustrate the theory through several families of quantum graphs. Based on joint work with Dawid Jasiński and Paweł Kasprzak (arXiv:2607.18473).