Alternate timeline(covers information from an alternate timeline)
In an alternate timeline in 2422, Morn's was the recreation facility on Deep Space 9, which was previously known as Quark's.
After the Klingons took over the control of Deep Space 9 and Starfleet left the station, Quark owned his own moon. Morn took over for him and renamed the bar years later.
Julian Bashir mentioned on a rescue mission for Benjamin Sisko in 2422 that Dax, Nog, Jake, and he could have a drink at Morn's after the mission was successful. (DS9: "The Visitor")