A 1:2 internally resonant mechanical system can undergo secondary Hopf (Neimark-Sacker) bifurcations, resulting in a quasi-periodic response when the system is subject to harmonic excitation. While these quasi-periodic orbits have been observed in practice, their bifurcations are not well studied, especially in high-dimensional mechanical systems. This is mainly because of the challenges associated with the computation and bifurcation detection of these quasi-periodic motions. Here we present a computational framework to address these challenges via reductions on spectral submanifolds, which transforms quasi-periodic orbits of high-dimensional systems as limit cycles of four-dimensional reduced-order models. We apply the proposed framework to analyze bifurcations of quasi-periodic orbits in several mechanical systems exhibiting 1:2 internal resonance, including a finite element model of a shallow-curved shell. We uncover local bifurcations such as period-doubling and saddle-node, as well as global bifurcations such as homoclinic connections, isolas, and simple bifurcations of quasi-periodic orbits. We also observe cascades of period-doubling bifurcations of quasi-periodic orbits that eventually result in chaotic motions, as well as the coexistence of chaotic and quasi-periodic attractors. These findings elucidate the complex bifurcation mechanism of quasi-periodic orbits in 1:2 internally resonant systems.