Atom emits photon when electron changes energy levels -- is this related to moving charge emitting e/m radiation?

Yes it is related. It is basically the same physics as the emission of radio waves from radio aerials. In both cases an oscillating charge emits a photon of a frequency equal to the frequency of oscillation.

In an atom it isn't obvious how there could be an oscillating charge, but there is. Explaining why an oscillating charge exists is a bit involved but if you fancy a challenge there is an excellent description of how this happen in Emilio Pisanty's answer to the question Is there oscillating charge in a hydrogen atom? Roger Barlow has linked the question How exactly does an electron falling back to its original state from an excited state produce electromagnetic waves? and his answer is also an excellent desription, though I think Emilio's is easier for the beginner.