Stochastic processes with absorbing states feature examples of non-equilibrium universal phenomena. While the classical regime has been thoroughly investigated in the past, relatively little is known about the behavior of these non-equilibrium systems in the presence of quantum fluctuations. Here we theoretically address such a scenario in an open quantum spin model which in its classical limit undergoes a directed percolation phase transition. By mapping the problem to a non-equilibrium field theory, we show that the introduction of quantum fluctuations stemming from coherent, rather than statistical, spin-flips alters the nature of the transition such that it becomes first-order. In the intermediate regime, where classical and quantum dynamics compete on equal terms, we highlight the presence of a bicritical point with universal features different from the directed percolation class in low dimension. We finally propose how this physics could be explored within gases of interacting atoms excited to Rydberg states.