= Solution
The set $\mathcal Q_n$ consists of <rooted planar map>[rooted planar maps] with $n$ faces, every face having degree four. The set $\mathcal Q_n^\bullet$ consists of these <planar quadrangulation>[quadrangulations] with an additional distinguished vertex.
For the <trivial bijection between planar maps and quadrangulations>, start from a rooted <planar map> with $n$ edges. Put a new vertex in every face and join it to the original vertex at every incident corner. Delete the original edges. The two endpoints of each deleted edge and the new vertices in its two adjacent faces bound a quadrangular face, so the result lies in $\mathcal Q_n$. Its bipartition distinguishes old from new vertices and reconstructs the original map. With the standard root convention this is a bijection, and hence
$$
\#\mathcal Q_n=\#\mathcal M_n.
$$
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