In this work, we propose and analyze an efficient numerical method for solving nonlinear Caputo q-fractional differential equations with non-smooth solutions. The numerical method is a linearized one and can be implemented on the graded mesh. For equations with fractional derivatives of order α∈(1,2), the method converges with second-order on uniform mesh. For α∈(0,1), second-order convergence can be obtained when the grading parameter of the nonuniform mesh is set appropriately.