does using a gold gear token activate the sivir battle mistress legend ability of "when you recycle a rune"?

No. Exhausting and killing a Gold gear token just adds one Power to your Rune Pool — it isn't a Recycle action, so Sivir's "when you recycle a rune" trigger never fires.

Gold's activated ability is "Kill this, [E]: [Reaction] — Add [A]," which generates Power rather than returning a rune to your Rune Deck (FAQ 9224). Recycling, by contrast, is specifically moving a card to the bottom of its deck — runes go to the Rune Deck (CR 417.1.b).

So Sivir only triggers off a real recycle event (for example, a basic Rune's "Recycle this: [Reaction] — Add [C]"). Notably, the Gold token Sivir makes is what you'd later cash in for Power — it doesn't loop back into triggering her.