No, Messenger doesn't have a built-in feature to detect if you take a screenshot of a story. So, you don't have to worry about it.
Generally, Messenger can't tell if you screenshot a story. It doesn't have that kind of monitoring functionality. However, it's always good to respect the privacy and terms of use of the app.
Messenger doesn't have the ability to know when you screenshot a story. This isn't a feature they've incorporated. Just use the app as intended and you'll be fine.