Here's what a quick search has found (on a German forum).
The Nice train is bookable to and from Austria, and the last stop is Innsbruck (going from the DB website, which unnervingly has two differing sets of times).
You might have to book through a specialist rail travel agency and not online.
However it's worth flagging up that the most scenic part of the journey over the Brenner Pass is done at night in both directions.
Maybe another member knows more? Or maybe the Man in Seat 61 might add it to his "to-do" list!