The short answer is: not yet, but soon.
Longer answer:
- APIs for Brackets extensions to add a new syntax/language are currently in progress, and will probably be done in about a month.
- Since Brackets uses the CodeMirror editor, syntax highlighting is driven by CodeMirror "modes." It doesn't appear that anyone has made a Twig mode yet. But it might not be too hard to build one, either based on the sample code for Mustache highlighting or the new "multiplexing mode" feature in CodeMirror 3 (Brackets is updating to CodeMirror 3 soon).
- Brackets may eventually support directly importing TextMate language bundles, like Sublime does, but that's a ways off.
As a stopgap, you could have Brackets just highlight Twig files as plain HTML -- better than no color coding at all. That'll be doable with the upcoming extension APIs, but if you're feeling adventurous you could hack your current copy of Brackets to do that right now. Just dig into the app folder, open www/editor/EditorUtils.js, and add the file extension after the case "html":
line.
(p.s. - I work on the Brackets core team. Thanks for giving Brackets a try!)