Can I prevent notifications from being forwarded to my watch during Do Not Disturb?

I have a Pixel 3 phone running Android 11, paired with a Garmin Venu watch. The Pixel can forward notifications to my watch via Bluetooth, with per-app notification controls specified in the Garmin connect app. I've noticed that whenever I set my phone on Do Not Disturb, I still get notifications forwarded to my watch. Is there a way on the phone to prevent notifications from being forwarded to my watch while in DND?