  We describe new $q$-deformations of the 3-dimensional Heisenberg algebra, the
simple Lie algebra $\mathfrak{sl}_2$ and the Witt algebra. They are constructed
through a realization as differential operators. These operators are related to
the modular group and $q$-deformed rational numbers defined by Morier-Genoud
and Ovsienko and lead to $q$-deformed M\"obius transformations acting on the
hyperbolic plane.
