The Angosian capital city in 2366
The Klingon First City in 2367.
A capital city, sometimes referred to as simply the capital, was where the seat of a planetary government and administrative center were located.
Built in the 14th century, Stratos was the tranquil capital city of the planet Ardana. (TOS: "The Cloud Minders")
In 2267, James T. Kirk planned to beam directly to the capital city of the Deneva colony following a distress call apparently sent by his sister-in-law Aurelan Kirk. (TOS: "Operation -- Annihilate!")
On Argelius II, the capital city was referred to as the chief city. In 2267, Hengist was the administrator for this city. (TOS: "Wolf in the Fold")
In 2268, the USS Enterprise was in synchronous orbit with the capital city of the planet Gideon, which caused Kirk some confusion when Odona seemed to have been beamed up instead of him down. (TOS: "The Mark of Gideon")
Paradise City was the capital of the planet Nimbus III in 2287. It housed the representatives of the three governments in control of Nimbus III at the time: the United Federation of Planets, the Romulan Star Empire and the Klingon Empire. (Star Trek V: The Final Frontier)
The Angosian capital city was a city on the planet Angosia III. The Angosian Senate operated under the Angosian Prime Minister from a building in the capital city. (TNG: "The Hunted")
The First City of the Klingon Empire was located on Qo'noS and was the location of the Great Hall, where the Klingon High Council met. (TNG: "Redemption")
In 2367, following William T. Riker's disappearance on Malcor III, Deanna Troi informed Mirasta Yale of his last known location, their capital city. Later, Data reported that they had scanned the area and still hadn't located him. (TNG: "First Contact")
According to the reference work Star Trek: Star Charts and the novels A Time for War, A Time for Peace and Articles of the Federation, Pike City is the capital city of the planet Cestus III.
Star Trek: Star Charts also names Tycho City as Earth's moon Luna's capital city.