We describe a general framework for the modeling and analysis of Markovian quantum antenna systems viewed as a special problem in quantum stochastic dynamics. The quantum radiator, which can be used to spatially direct radiated quantum states in future quantum communication systems, is modeled as an open quantum system capable of controlling the composition of its radiation modes through external source manipulations while in continuous interaction with the surrounding thermal and random environment. Our analysis and computational method are based on deploying the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation to account for environment-induced jump processes such as quantum dissipation and decoherence. We construct a definition of the quantum antenna directivity inspired by the corresponding formulas in classical antenna theory and use the GKSL formalism to derive several versions of the quantum directivity formula. As a computational example, we study the flow of the density operator of a coupled two-level quantum dot (qubit) array, excited by classical external signals with variable amplitude and phase, which is evolved in time using the quantum Liouville-type equation (the master equation). It is shown that by manipulating the amplitude and phase excitations of individual quantum dots, one may significantly enhance the directive radiation properties of the Markovian quantum antenna system.