Zalkon was an inhabited planet in an unknown star system in Sector 9569. This was the homeworld of the warp-capable humanoid Zalkonians. The Zeta Gelis star cluster, a star cluster in a neighboring sector, was located 3.2 parsecs from Zalkon.
In 2366, John Doe and three other Zalkonians fled their homeworld in order to undergo the complete metamorphosis of their species from corporeal to non-corporeal beings. All but John Doe were hunted down by Zalkonian authorities before they could finish the transformation, however.
When the crew of the USS Enterprise-D found a Zalkonian escape pod on a planet in the Zeta Gelis star cluster, with a badly wounded John Doe on board, they were able to reconstruct the course of his ship, back to Zalkon. At the time, Zalkon was on the Enterprise-D's intended course, and was already projected to be reached within 3 weeks, during their star charting mission. (TNG: "Transfigurations")
This planet was only mentioned in dialogue.
According to Star Trek: Star Charts ("United Federation of Planets II") and Stellar Cartography: The Starfleet Reference Library ("Stellar Cartography", pp. 18-19; "Federation Historical Highlights, 2161-2385”), Zalkon was located in the Zalkon system. This system was located in the Beta Quadrant.