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Here is an old thread where a more difficult question is addressed: mathoverflow.net/questions/103111/…. The thread has also the sketch of an answer by @Anthony Quas (mathoverflow.net/a/103127/16537), but there is no reference to the question stated in the OP there.
@Gabriel. I've at least two issues with your answer. First, as you too have remarked, it doesn't answer my question. Second, I'm not so convinced that, after filling in the technical details that you're alluding to, the construction you are suggesting will still be so fairly easy: I might mention a number of situations where constructions involving densities are relatively easy (or even trivial to some degree) as long as the "relevant parameters" are rational, but get significantly more complicated otherwise. In any case, thanks for your contribution to the discussion.
For what it's worth, here is a proof that the asymptotic density of $Q_2$ is zero, which doesn't depend on Landau's result: mathoverflow.net/a/205866/16537.
