Can my opponent reach to me playing a Spinning Axe and attaching it to an Over Zealous Fan with a Gust?

Yes, your opponent can play Gust in response to your Spinning Axe.

Here's the sequence:

1. You play Spinning Axe (via Quick-Draw, which gives it [Reaction]) targeting your Overzealous Fan to attach. This puts Spinning Axe on the chain.
2. The state becomes Closed. Your opponent responds with Gust (also [Reaction]) targeting your Overzealous Fan. Gust goes on the chain on top.
3. Resolving LIFO: Gust resolves first — your Overzealous Fan (2 Might ≤ 3) is returned to your hand.
4. Spinning Axe then resolves — it tries to attach to the Fan, but the Fan is no longer on the board, so the attachment instruction is ignored and the Axe enters the battlefield unattached (or is recalled during the next cleanup).

This is the same interaction covered in FAQ #9813 — a Reaction spell like Gust can always be played in response to a chain item, and it will resolve before the item it's responding to.