Quote by Gza
There is a person I know that believes that there is a nonzero possiblity that the moon will suddenly "jump" to a different spatial location, based on the axioms of quantum mechanics.

The propbability is nonzero by QM, but is so small that it would take probably many and many times the age of the universe to become a reasonable chance to make such a leap (even a mm probably).
Quote by Gza
Another misunderstood idea is that quantum motion is inherently different than "classical motion" as if an electron making a jump from a certain energy state to another is different than the moons orbit.

It is actually quite different, maybe not in principle but at least to many degrees of accuracy. For the many particles of which the moon consists, the quantum behaviuor can be safely neglected and we can speak of it's position and followed path and orbit.
Quote by Gza
The main point to understand here is that we can not consider any particle as a single entity, this is in fact demanded by QM, instead we can only view every particle of the moon as part of a system which indeed does have 0 probability as a whole to move anywhere but along the path described by general relativity.

In principle still nonzero, only unimaginably small.
If you (by followoing this persons argument) you would consider the moon as one entity with a single wavefunction (which is quite reasonable) with this wavefunction a huge mass is associated. By Heisenbergs uncertainty principle there is a very small uncertainty in the the verlocity and position of the moon. In a formula
[tex]\Delta x \Delta v < \hbar/2m [/tex]
So the product of uncertainties in velocity and position of the moon is smaller than the unimaginably small number on the right. For all purposes the unvcertainty in position (and thus the chance of the moon leaping to some different position) is negligible.