Kirk did not have a ”thing.” Nor did Sisko or Janeway. The idea that everyone likely to give the command to go to warp needs to have their own catchphrase is unsustainable.
I’ll give you Kirk and Sisko, but Janeway absolutely did. Whereas Picard had “engage” and “make it so,” Janeway had “Do it.” It wasn’t always her ‘warp’ thing, but it was certainly her go to phrase for finishing an order.