The development of quantum information processing and quantum computation goes hand in hand with the ability of addressing and manipulating quantum systems. Quantum Control Theory has provided a successful framework, both theoretical and experimental, to design and develop the control of such systems. In particular, for finite dimensional quantum systems or finite dimensional approximations to them. The theory for infinite dimensional systems is much less developed. In this talk I propose a scheme of infinite dimensional quantum control on quantum graphs based on interacting with the system by changing the self-adjoint boundary conditions. I will show the existence of solutions of the time-dependent Schrödinger equation, the stability of the solutions and the (approximate) controllability of the state of a quantum system by modifying the boundary conditions on generic quantum graphs.