Well at least that's one thing we agree on. I was going to assume that at some level this would be a new, as yet undiscovered, interaction between matter. Probably, since it's clearly not an interaction at a fundamental level (we'd have seen it already otherwise), then it would be something that appeared only between, say, brains as a whole (or the mind?).
Then you could model this in one of two ways. In both cases the information would be passed by a new form of energy -- and this energy could be quantised (although I may try to relax that assumption).
Anything beyond this I haven't got a clue yet. I did an initial calculation earlier assuming that you needed to be close to someone in order to be able to "read" them that suggested a remarkably high probability of such psychic feelings occurring, although that was almost literally back-of-the-envelope.