I hope that the crew uses rank or 'Mister' when addressing each other. I do not believe that they all know each other well enough yet not to. The exception being (maybe) Kirk & McCoy. And only when speaking in private. Calling the Captain 'Jim' on the bridge is inappropriate. IMO.
Assuming you're talking about Into Darkness
, from what I remember of the opening 9-minutes Sulu and Uhura called each other "Sulu" and "Uhura", with Kirk using "Uhura", "Scotty" and "Bones". Only Spock invoked rank.
This Trek has even less of an emphasis on ranks than the old. All are trained, one gets to be the captain, thats about it.