You can always make a sheet number customproperty which can be linked to the bom but as far as filling it in automatically, I don't think there is a built in option for that. Maybe someone can chime in with a macro if it is possible.
You might be able to get away with using flag notes and their zone location (which also references sheet number). I am not sure if it will work, but you could give it a try.
I don't know how to link it to a BOM and I am not sure if a balloon from a flag note will convert to a balloon from a BOM.
But, you might be able to get away with a Numbered set of notes that links a flag note to each balloon. Would be some work, but would update for you (I think).