Nope! Wolfram Alpha puts the orbital kinetic energy of the Moon at 3.425e28 joules, or 8.186e18 tons of TNT [1]. The Tsar Bomba, the biggest nuclear weapon ever detonated, had the energy equivalent of 50 megatons of TNT (5e7 tons), which is 100 billion times smaller than the Moon's kinetic energy. The Moon is just really heavy - it's unlikely we could ever do much to change it's orbit (with current technology at least!).
What if we build a gigaton-range warhead (which modern rockets could easily lob[1]) and watch for an opportunity to deflect a comet or large asteroid? The parts I have no idea about are: how often suitable projectiles come close enough to the Moon's path that a Gt nudge would do the deed; and whether we could predict the effects of a huge nuke well enough to steer the object with the requisite precision--but it seems to me interfering with the Moon is only a bit more difficult than defending against a doomsday impact on Earth, so I hope it would be possible.
I can't currently find a link, but I've seen speculation that nuclear explosions on the moon could serve to eject some of the moon's mass. Even a small change would necessarily change the moon's orbit very slightly – and over enough time, the position of Earth-moon system would be quite different from what it would have otherwise been.
(This was proposed as a way to move the Earth out of the path of an approaching body, given enough warning, without actually having to reach the much-further-away approaching body, and alter its path.)
[1] - https://www.wolframalpha.com/input/?i=kinetic+energy+of+the+...