The goal of this paper is to investigate, by applying the standard Caputo fractional $q$-derivative of order $\alpha$, the existence of solutions for the singular fractional $q$-integro-differential equation $D_q^\alpha x(t) = w\big(t , x(t), x'(t), D_q^\beta x(t)$, $\int_0^t f(r) y(r) dr \big)$ under some boundary conditions where $w(t, x_1, x_2, x_3, x_4)$ is singular at some point $t$ belongs to $[0,1]$. We consider compact map and using Lebesgue dominated theorem for finding solutions of the problems. besides, we present results whenever the function is completely continuous. Lastly, we present some examples illustrating the primary effects.
