TheUSS Trieste was a Merced class escort serving in Starfleet during the 14th century.
By 2364 she was serving on the Klingon/Federation border as part of the Second Fleet and would continue to do so for over a decade.
In 2372 the vessel responded to a distress call from the runabout Cau Lau after it had been attacked by a Klingon Bird of Prey. ("Roadtrip")
Data had served on the ship prior to the USS Enterprise-D.