A new family of fractional counting processes based on a three-parameter generalized Mittag-Leffler function was introduced and studied. As applications we develop a fractional generalized compound process, introduce and develop fractional generalized Bell polynomials and numbers, fractional generalized Stirling numbers of the second kind, and a new family of quantum coherent states. Stretched quantum coherent states, which are a generalization of the famous Schrödinger-Glauber coherent states, were also introduced and studied. In particular cases, the presented results reproduce known equations for Poisson and fractional Poisson probability distributions, Bell numbers and fractional Bell numbers, Stirling numbers and fractional Stirling numbers of the second kind, as well as for known quantum coherent states.