Mode-exchange collisions occur in multi-mode Bose-Einstein condensates, in some cases, producing particle loss in the system. Nevertheless, little is known about mode-exchange collisions in the case when particles do not escape the trap. We show that these processes are relevant in quantum properties of the system such as the evolution of the relative population, the self-trapping effect and the probability distribution of particles. Surprisingly, including mode-exchange terms in the model of the two-mode condensate allows for an exact analytical solution. Using this solution, we show that collisions favor the generation of entanglement between the modes of the condensate.