In present study, we investigate the existence of solution for a $k$--dimensional system of multi-term fractional $q$--integro-differential equation with special boundary conditions under some different conditions. In this way, we provide different results for the existence of solutions for the system and also for obtaining unique solution for the system under different conditions. We also present three numerical examples in which by using the Legendre multi-wavelet method, we approximate solutions of the system. These examples illustrate our main results.
