First Hospital was a 24th century medical facility, the largest hospital in First City, the capital city of the Klingon Empire on the planet Qo'noS.
History and specifics
In 2375, After Teilani was poisoned, her child Joseph Kirk was treated and kept in stasis at the facility's pediatric center. Leonard McCoy and Andrea M'Benga spent a month there, long enough to have set up a temporary office. Romulan genetics experts Prein and Troltan were expected to arrive three weeks later, and looked forward to examining a child of his unique heritage. James T. Kirk was later beamed away from the hospital by Tiberius. (TOS novel: Preserver)