The Trieste was the nearest starship to Starbase 74 in 2364, when the USS Enterprise-D was hijacked from that facility. It was suggested that the Trieste might be able to intercept the Enterprise-D. Unfortunately, the Trieste was determined to be too small and too slow to render assistance, on top of the fact that it was sixty-six hours away at the time. (TNG: "11001001")
In 2368, the Trieste joined Captain Picard's fleet that blockaded the Klingon-Romulan border during the Klingon Civil War. The ship was one of twenty-three assembled for the fleet and was one of the seventeen chosen for the tachyon network. Identified by its name and registry number, the ship's position in the network was monitored from the Enterprise-D.
When the network was disrupted by a high-energy burst released by the Romulans, the net was no longer effective in a radius of ten million kilometers around the USS Sutherland. Subsequently, Picard ordered the fleet to proceed to Gamma Eridon at maximum warp, where the fleet would attempt to reestablish the net before the Romulans got there. (TNG: "Redemption II", okudagram)
In 2369, the Trieste was on a pulsar research project. The ship was named on the chart "Starfleet Operations-Sectors 21538-23079" in the Enterprise-D observation lounge. (TNG: "Chain of Command, Part I" okudagram)
During one of the Star Trek: Deep Space Nine kitbash rounds, a model of this ship was constructed. The ship was given the registry number NCC-3724, being only one digit short of the established registry. Several Polaroids, provided by Dan Curry to the Flare Sci-Fi forums, were taken of this ship, along with the USS Antares and the Excelsior-class USS Excel (NCC-2020).  There is no evidence that this model appeared in an episode.
The Trieste was named for the bathyscaphe in which oceanographer Jacques Piccard (for whom Captain Picard was named) explored Earth's Marianas Trench in the 1960s. (Star Trek Encyclopedia (4th ed., vol. 2, p. 419))