If Joe envied something, would it be proper to call that something "Joe's Envy"? Considering the usage in a team name but I am not sure if it makes sense.
No. Envy is an emotion. To envy is to experience that emotion in relation to something. The object of envy is not envy.
However, from a literary perspective, you could still use "Joe's Envy" as a story or book title.
Just not as the object of that envy.