It might be feasible if there were a source of fuel on the moon, like some sort of automated refinery. Although that would require a Titan-like moon, which probably aren't very common.
A much more practical plan would be to simply find an asteroid with a high velocity relative to the Hydral planet and nudge it into a collision course. That's not nearly as impressive-sounding as moving a moon, though, so I guess
Rule of Cool is in effect.
EDIT: Now that I think about it some more, the moon would also have to be small enough that the reaction mass from the rockets would clear orbit, or else they wouldn't move the moon at all. And a moon that small wouldn't seem likely to have the necessary fuels, so yeah.