We derive detailed and integral fluctuation relations as well as a thermodynamic uncertainty relation constraining the exchange statistics of an arbitrary number of noncommuting conserved quantities among two quantum systems in transport setups arbitrary far from equilibrium. These universal relations, valid without the need of any efficacy parameter, extend the well-known heat exchange fluctuation theorems for energy and particle transport to the case of non-Abelian quantum transport, where the noncommutativity of the charges allows going beyond standard thermodynamic behavior. In particular, we show that this can lead to enhanced precision in the current fluctuations, and it allows for the inversion of all currents against their affinity biases.