If it doesn’t what does it do?
I refer to a phone notifying you of a call - with a ring or otherwise - as “going off.” However, after looking this up, that apparently means “to explode” (possibly metaphorically, depending on which source you trust).
So far, even on vibrate or silent, my phone has not done that.
I also say “your phone is going off” as it can refer to either the alarm, incoming call, or other notifications. It’s definitely an accepted alternative meaning for the phrase.
“Going off” can also mean “be in the process of spoiling / going bad” when referring to foodstuffs, which I only mention to point out that the phrase has at least one alternative use beyond exploding, even if alarms and such aren’t considered. More usually in the past tense of “gone off”, but it gets some use in the present tense.