This paper presents an online kinodynamic motion planning algorithmic framework using asymptotically optimal rapidly-exploring random tree (RRT*) and continuous-time Q-learning, which we term as RRT-Q. We formulate a model-free Q-based advantage function and we utilize integral reinforcement learning to develop tuning laws for the online approximation of the optimal cost and the optimal policy of continuous-time linear systems. Moreover, we provide rigorous Lyapunov-based proofs for the stability of the equilibrium point, which results in asymptotic convergence properties. A terminal state evaluation procedure is introduced to facilitate the online implementation. We propose a static obstacle augmentation and a local replanning framework, which are based on topological connectedness, to locally recompute the robot's path and ensure collision-free navigation. We perform simulations and a qualitative comparison to evaluate the efficacy of the proposed methodology.