No Starfleet in early TOS season one. They were work shopping names like crazy.
Actually, the term Starfleet shows up early in season 1.
https://memory-alpha.fandom.com/wiki/Starfleet?mobile-app=false
The first episode whose production incorporated the terms "Starfleet" and "Starfleet Command" was TOS: "Court Martial". However, the final draft and a revised final draft of that episode's script (dated 26 and 29 September 1966 respectively) used the term "Galactic Command" instead of "Starfleet Command". By comparison, the term "Starfleet" was extensively used in the final draft script of "The Menagerie" (issued in October 1966). In terms of air date too, that name as well as "Starfleet Command" debuted in "The Menagerie, Part I" (which aired 17 November 1966), though the term "Star Service" and "Spacefleet Command" were later used in "The Conscience of the King" (broadcast on 29 September 1966) and "The Squire of Gothos" (aired 12 January 1967) respectively, prior to the initial airing of "Court Martial" (on 2 February 1967). Another early variant of the organization's name was "Space Central" in "Miri", which preceded all those episodes in both airing and production order (broadcast on 27 October 1966).