Vernal Galaxy was a region of the Milky Way Galaxy, the third quadrant of which contained the Talos star group. (TOS: "The Menagerie, Part I")
The term "vernal" (a dated term meaning springlike or related to spring), is used in astronomical coordinate systems to refer to the point or region of sky defined by the vernal equinox. Its nature as a section of the galaxy rather then a different galaxy was made clear by a statement from Captain Christopher Pike in "The Cage", where while on Talos he claimed he was "from a stellar group at the other end of this galaxy."
Talos IV was said to lay in the Alpha Quadrant in "If Memory Serves", suggesting that so did at least part of Vernal Galaxy. Starbase 11, located just two light years from Talos IV, might also be in Vernal Galaxy.
The term "Vernal Galaxy" in the official Starfleet Command report was used in a sentence without a "the" preceding it, and was written completely in lowercase letters despite several important words being written in all caps.