Panama City was a city on Earth, located in Florida, which was part of the United States of America. The city was located in the northwest corner of the state, on the coast of the Gulf of Mexico.
Earth Starfleet officer, Charles Tucker III, was born in Panama City in 2117. Trip grew up in and attended school in Panama City. His parents continued to live in Panama City until 2153 when the Xindi attacked Earth. After the attack his parents moved to Mississippi. (ENT episodes: "Fusion", "The Augments"; ENT novel: Last Full Measure)
In one alternate timeline, Trip Tucker and his wife, T'Pol lived in Panama City from sometime after 2161 until 2199. (Myriad Universes - Infinity's Prism novel: A Less Perfect Union)