A goose (plural: geese) was a type of aquatic fowl indigenous to Earth.
A flock of geese flew across the sky of the Shore Leave Planet when Captain Kirk visited the planet in 2267. (TOS: "Shore Leave")
While the USS Enterprise attempted to escape the USS Reliant by entering the Mutara Nebula, Saavik noted that static discharge and gas clouds of the nebula would disable the ships' visuals and shields. Spock noted how her observations were actually to their benefit by replying, "what's sauce for the goose, Mr. Saavik. The odds will be even." (Star Trek II: The Wrath of Khan)
Data once stated he was possibly "pursuing an untamed ornithoid without cause", meaning on a wild goose chase. (TNG: "Data's Day") Spock was also once accused of taking the USS Enterprise on a wild goose chase, to which he replied that he was following logic and not a wild aquatic fowl. (TOS: "The Gamesters of Triskelion") Later, as the crew struggled with the M-5 multitronic unit's takeover of the ship's systems, Spock recalled something of the expression and stated "we've been doing what used to be called pursuing a wild goose." (TOS: "The Ultimate Computer")
In 1996, after Rain Robinson discovered a UFO in Earth's orbit, Henry Starling attempted to downplay the discovery by noting that it was probably a flock of geese. Robinson disagreed, as flocks of geese don't park themselves 20,000 kilometers in orbit above Earth. (VOY: "Future's End")