The Ajax was one of the ships that received Kosinski's experimental warp drive upgrades in 2364. The engineering staff reported a measurable increase in performance and engine efficiency following the upgrade. It was later suggested that these enhancements in performance may have been attributed to straightening out simple engine inefficiencies experienced by the older ship. (TNG: "Where No One Has Gone Before")
In 2367, the Ajax was on a deep space exploration mission in Sector 21834. The ship was named on the chart "Starfleet Operations-Sectors 21166-23079" on the USS Enterprise-D bridge. (TNG: "Brothers" okudagram)
In 2368, the Ajax 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 USS 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 Ajax was on a deep space exploration mission in Sector 21396. 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)
In the episode "Brothers", the Ajax was identified as an Excelsior-class starship with the registry of NCC-13554.
Soon after the episode was taped in the summer of 1990, the class and registry were retconned for an internal list of starships, dated September 26, 1990. In future episodes, the registry was mentioned in a tachyon detection grid seen in "Redemption, II" and the class was mentioned in a chart seen in "The Chain of Command, Part I". 
In Star Trek: Legacy, Ajax appeared as a playable starship, designated as a scout.