If it does what does it fall under?

The Stargazer-class is a Heavy Cruiser in-game. However, if you're refering to the U.S.S. Stargazer that Picard was captain of before the Enterprise-D then no, it's not in-game as that ship was a Constellation-class ship.

The closest approximation is probably the Cheyenne Class. It's not the same, but is probably the closest you can come to it in game. All classes of cruiser at this level (except the Excelsior) appear to be based off of the Constellation class that the Stargazer belonged to.

So the one that Picard pilot was another class of Starship? The one with the four warp nacelle?

Yes

The USS Stargazer, Jean Luc Picard's command before the Enterprise was a Constellation-Class cruiser.

The Stargazer-Class in game is named for Picard's stargazer but is not of the same class. The Constellation class is not in the game at this time.