In　order　to　overcome　the　complexity　of　the　theoretical　analysis　caused　by　using　decomposition　method　to　explore　the　finite-time　synchronization　behavior　of　fractional-order　quaternion-valued　neural　networks　(FOQVNNs),　we　aim　to　deal　with　this　problem　directly　instead　of　decomposition.　Firstly,　two　inequalities　about　quaternion　are　developed　to　broaden　the　current　achievements　in　quaternion　field.　Secondly,　a　fractional　differential　inequality　is　established　by　using　Laplace　transform　and　applying　the　definition　of　Mittag-Leffler　function.　Then,　by　employing　the　presented　inequalities　and　two　different　quaternion　control　strategies,　some　new　conditions　are　derived　to　guarantee　the　finite-time　synchronization　of　the　delayed　FOQVNNs.　Finally,　two　numerical　examples　are　given　to　illustrate　the　correctness　of　the　main　results.