The first is fine. The second is unidiomatic.Am I right here?
"Thank God your alarm clock went off. We didn't miss the bus." - the alarm clock started to ring, it is fine.
"To bad your alarm clock went off. You need to get a new one." - the alarm clock is broken.
Also:Three different meanings of "go off".
1. To do what it is supposed to do - an alarm goes off (it rings), a bomb goes off (it explodes).
2. To have switched to its "off" state - a heater goes off - it stops heating, a light goes off - the room is now dark.
3. To become putrid - this meat has gone off - it stinks.