First, note that it's Brokk's brother who is clearly the master smith and does the actual smithing, while Brokk works the bellows. This brother is given as either Eitri or Sindri in different manuscripts of Snorri's Edda. The name Sindri also appears in Völuspa, where it is told he had a hall in Nithavellir ("the Dark fields").
The forge in itself goes unnamed in the story (it's short, you can read it in a few minutes. See e.g. here). Furthermore, I can't recall any forge that are given a name in Norse mythology. It would seem that forges were not in themselves considered that important, and certainly not when compared to the skill of the smith, which is emphasized several times in several completely different tales.