Given a pair of coordinates $(x,y)$ in 2D, find the points inside the circle $C((x,y),R)$
Obviously not. You must inspect all the points and, on a classical, deterministic computer, you can't possibly look at $n$ points in only $O(\log n)$ time.

Note that this answer applies equally to the original question (is it possible in $\log n$ time?) and the revised one (is it possible in $|Q|+\log n$ time?). The answer could well be $Q=\emptyset$, but you can't figure that out in time $o(n)$.