When someone mentions @username in their Twitter bio, does the person get notified about this i.e their @username is mentioned in someone's bio?
I just tested it with two of my accounts.
When I added an @mention of one of them in the profile of the other, while it made a nice link to the profile of that other account, I did not receive any sort of notification about it. Nothing in "interactions" or "mentions", and certainly nothing by email.
So the answer to your question would appear to be No.