In this paper, we study a class of fractional $q$-differential inclusion of order $0 < q <1$ under $L^1$-Caratheodory with convex-compact valued properties on multifunctions. By the use of existence of fixed point for closed valued contractive multifunction on a complete metric space which has been proved by Covitz and Nadler, we provide the existence of solutions for the inclusion problem via some conditions. Also, we give a couple of examples to elaborate our results and to present the obtained results by some numerical computations.