All the info I could find states the following (however, they could have been awarded more unit commendations and I just wasn't looking in the right places). I hope this helps.
CentCom recieved 7 Joint Meritorious Unit Awards between 1985 and 2000. The last one was actually awarded in March 2001, but it covers all of CentCom's activities between November 1999 and March 2001.
SOCCent recieved 4 Joint Meritorious Unit Awards between 1985 and 2000. Again, their fourth award covers SOCCent's activites between November 1999 and March 2001.
MTV Video Music Awards 1985 - 1985 TV was released on: USA: 13 September 1985
Imagen Awards was created in 1985.
TEC Awards was created in 1985.
The 27th Annual Grammy Awards - 1985 TV was released on: USA: February 1985
The 39th Annual Tony Awards - 1985 TV was released on: USA: 2 June 1985
The 57th Annual Academy Awards - 1985 TV was released on: USA: 25 March 1985
World Habitat Awards was created in 1985.
The 6th Annual Black Achievement Awards - 1985 TV was released on: USA: 11 January 1985
The 45th Annual Golden Apple Awards - 1985 TV was released on: USA: 8 December 1985 (Beverly Hills, California)
The 1th Annual Commitment to Life Awards - 1985 TV was released on: USA: 19 September 1985 (Los Angeles, California)
The cast of The National Association of Theater Owners Awards - 1985 includes: Danny DeVito as Himself - Winner Clint Eastwood as Himself - Winner Jeff Goldblum as himself Kathleen Turner as Herself - Winner
You are probably thinking of "Relax" which was their only number one hit. But it was before 1985, but they won awards for it in 1985.