  We give enumerative interpretations of the polynomials arising as numerators
and denominators of the $q$-deformed rational numbers introduced by
Morier-Genoud and Ovsienko. The considered polynomials are quantum analogues of
the classical continuants and of their cyclically invariant versions called
rotundi. The combinatorial models involve triangulations of polygons and
annuli. We prove that the quantum continuants are the coarea-generating
functions of paths in a triangulated polygon and that the quantum rotundi are
the (co)area-generating functions of closed loops on a triangulated annulus.
