The fictitious US ship here is USS Monroe Smith (FFG-62 or FFG-63). The crewmembers wear baseball caps bearing the hull number FFG-63, but a plaque at the ship's bell bears the number FFG-62. The fictitious ship may or may not be named for David Monroe Smith, an enlisted soldier of the US Army, who died in action in 1950 during the Korean War, and who posthumously received the Medal of Honor for his "conspicuous gallantry and outstanding courage" -- that is, for having flung himself upon an enemy grenade, thereby sacrificing himself but saving five fellow infantrymen. In the US Navy there is no frigate bearing the number of 62 or 63. The one with the highest number is USS Ingraham (FFG-61).
The footage used for the establishment shots near the start shows three different ships of the US Navy -- representing the US frigate, the Turkish destroyer, and the Greek destroyer escort. Respectively, they are USS John Moore (FFG-19), of the Oliver Hazard Perry class, named for a submarine skipper who died in the sinking of his ship in action against the Japanese Navy, USS Carpenter (DD-825), of the Gearing class, named for a Naval aviator, and USS Robert Peary (DE-1073), of the Knox class, named for the rear admiral who is generally regarded as the first person who, in 1903, reached the North Pole.