Contact processes play an important role in classical non-equilibrium dynamics, describing the spreading of diseases, the dynamics of earthquakes and forest fires, and the distribution of information through the internet. Here we show that their quantum counterpart, where the spreading occurs through coherent couplings, displays even richer dynamics and offers new means of control. A quantum contact process on a topologically non-trivial one-dimensional lattice can be confined to a protected subspace corresponding to either a single site or a fully excited lattice. Furthermore, excitation spreading can be controlled to occur in quantized steps and on demand when employing topological pumps. We show that the many-body dynamics of excited domains can be mapped to an effective single-particle model, which also determines the topological properties. Throughout this work, we consider a specific type of contact process corresponding to coherent Rydberg facilitation in a tweezer array of trapped atoms in a one-dimensional lattice.