Color centers in diamond have emerged as leading solid-state artificial atoms for a range of quantum technologies, from quantum sensing to quantum networks. Concerted research activities are now underway to identify new color centers that combine stable spin and optical properties of the nitrogen vacancy (NV⁻) with the spectral stability of the silicon vacancy (SiV⁻) centers in diamond, with recent research identifying other group IV color centers with superior properties. In this paper, we investigate a class of diamond quantum emitters from first principles, the group III color centers, which we show to be thermodynamically stable in a spin-1, electric-field insensitive structure. From ab initio electronic structure methods, we characterize the product Jahn-Teller (pJT) effect present in the excited state manifold of these group III color centers, where we capture symmetry-breaking distortions associated with strong electron-phonon coupling. These predictions can guide experimental identification of group III vacancy centers and their use in applications in quantum information science and technology.