We show that in the model obtained by iteratively pseudo-intersecting a Ramsey ultrafilter via a length-ω₂ countable support iteration of restricted Mathias forcing over a ground model satisfying CH, there is a unique Q-point up to isomorphism. In particular, it is consistent that there is only one Q-point while there are 2^c-many near-coherence classes of ultrafilters.
Halbeısen et al. (Fri,) studied this question.