In this manuscript, by using the Caputo type and the Riemann-Liouville type fractional $q$-derivative, we investigate the coupled system of fractional $q$-differential equations ${}^c\mathcal{D}_q^{\beta_1} y(t)=f_1 (t, y(t), z(t))$, for $t\in [0,l]$ and $i=1,2$ under integral boundary conditions. In fact, we give some results by considering Leray-Schauder alternative theorem. Lastly, we give an example, some related algorithms and numerical results.
