This paper investigates the global Mittag-Leffler synchronization (GMLS) issue for fractional-order delayed quaternion-valued neural networks, which include leakage delays and transmission delays. First, a new lemma is established to estimate the Caputo fractional derivative for quaternion self-conjugate quadratic Lyapunov function, and a novel quaternion-valued linear feedback controller is designed. Then by utilizing quaternion matrix theory and our proposed lemma, some succinct GMLS criteria are derived respectively in quaternion-valued linear matrix inequality (LMI) form and complex-valued LMI form, which simplify and extend some previous work on the synchronization control for quaternion-value neural networks. Finally, a numerical example is given to validate our theoretical results.