Chern numbers characterize topological phases in a wide array of physical systems. However, the resilience of system topology to external perturbations makes it challenging experimentally to investigate transitions between different phases. In this study, we demonstrate the transitions of a Chern number from 0 to 3, synthesized in an electronic-nuclear spin system associated with the nitrogen-vacancy (NV) centre in diamond. The Chern number is characterized by the number of degeneracies enclosed in a control Hamiltonian parameter sphere. Topological transitions between different phases are realized by varying the radius and offset of the sphere such that the Chern number changes. We show that the measured topological phase diagram is consistent with numerical calculations and can also be mapped onto an interacting three-qubit system. The NV system may also allow access to even higher Chern numbers, which could be applied to exploring exotic topology or topological quantum information.