In this study, we use certain mathematical tools to analyze the solutions of a system of fractional $q$-differential equation ${}^C\mathbb{D}_{q}^{\sigma_i} [\wp](\mathfrak{t}) = \mathfrak{w}_i( \mathfrak{t}, \wp(\mathfrak{t}), {}^C\mathbb{D}_{q}^{{}_i\nu_{j}} [\wp](\mathfrak{t}), \mathbb{I}_{q}^{{}_{i}\nu_j} [\wp] (\mathfrak{t}) )$, $i=~1$ whenever $\mathfrak{t} \in [0, \mathfrak{t}_0]$, and $i=2$ whenever $\mathfrak{t} \in [\mathfrak{t}_0, 1]$, for $j=1,2$, such as fixed point theorem of Krasnoselskii and Banach contraction principle, under simultaneous Dirichlet boundary conditions. Here, we use standard definitions of the Liouville-Caputo fractional type $q-$derivative and Riemann-Liouville $q-$integral. Some illustrative examples with numerical results are discussed, too.
