Can you use a quick draw gear after both passing focus and before assigning combat damage?

No. Once both players pass focus in sequence, the Showdown closes and combat proceeds directly to the damage step — there is no priority window to play anything, including a Quick-Draw gear at Reaction speed.

Quick-Draw only grants Reaction timing (Rule 819); it doesn't create a new window where none exists. Since both players passed on an empty chain, no player has Focus or Priority when the Showdown closes, so the gear can't be played before damage assignment (Rules 347.2.a, 348.1; FAQ #6389, #7747).

If your opponent plays anything on the chain before passing, focus returns to you and you'd get your reaction window then — but once both pass, the window is gone.