This paper deals with the existence of nonnegative solutions for a class of boundary value problems of fractional q-differential equation Dqσc[k](t)=w(t,k(t),cDqζ[k](t)) with three-point conditions for t∈ (0 , 1) on a time scale Tt0={t:t=t0qn}∪{0}, where n∈ N, t∈ R, and 0 < q< 1 , based on the Leray–Schauder nonlinear alternative and Guo–Krasnoselskii theorem. Moreover, we discuss the existence of nonnegative solutions. Examples involving algorithms and illustrated graphs are presented to demonstrate the validity of our theoretical findings.