Hi @CritterCrew please allow me to give you a warm welcome to the Ubisoft Forums! I'm glad to help you, and I can empathize with your annoyance of the "New Quest" alert never disappearing! This issue was previously report, but confirmed as fixed in a previous patch. However, if you're having it currently on the 1.6.0 title update, we can look into investigating this again. To continue, please try verifying the game files in Ubisoft Connect to see if there's any improvement.
Should the problem continue, please capture a video of you going through your quest log, including the completed quests. Also, please show you navigating to the game home screen to show the current build number. When you've got that recorded, please upload it to either YouTube (as an unlisted or public video) or Twitter. You can then send us a link via this thread.
Thanks for working with me on this! Stay strong, Viking!