If in response to my opponent playing Gust on their unit I respond with Rengar Trophy Hunter to that battlefield does a showdown occur?

Yes, a showdown does occur — but it will be a non-combat showdown (no fighting), and you will likely conquer the battlefield.

Here's the full sequence:

1. Rengar enters while Gust is still on the Chain. Since you don't control that battlefield and it isn't already contested, Contested status is applied (Rule 187.3.a.1).
2. Gust resolves, bouncing the opponent's unit to hand. Only Rengar remains at the battlefield.
3. Chain empties → Cleanup occurs. The Cleanup stages a Showdown at that battlefield (Rule 323.8). Since only your unit remains, no Combat is staged (Rule 323.9).
4. The Turn Player initiates the showdown. With only Rengar present, you establish Control (Rule 348.2.a). If you haven't scored that battlefield this turn, you Conquer and gain a point.

Key distinction from surprise-defense scenarios: When Rengar enters in response to a move, both units stay and a Combat showdown results. Here, Gust removes the opponent's unit before the showdown begins, so you win control without combat. Rengar's unique text lets him Ambush to enemy-occupied battlefields even without your own units there — that permission works here since an enemy unit was present when he was played.