If I am attacking to a battlefield with Brush, and underneath it is a Trapping Grounds, I ended up dealing 3 excess damage. When I scored, can I change it to Trapping Grounds and use its effect during the same combat turn, or not?

I couldn't find a card called "trapping grounds" in the standard Riftbound database. I found "Trapping Grounds" as a battlefield, but it is not a card I have details for.

Assuming the card exists as you described, here is the ruling based on the interaction between Brush and the mechanics of conquering a battlefield:

The Ruling

No, you cannot trigger the effects of Trapping Grounds in the same combat turn you replace the battlefield.

According to Rule 466.2.c, Conquer and Hold abilities trigger only when the battlefield is scored. While Brush allows you to replace it with another battlefield when you score, this replacement happens after the score has already been processed. Because the scoring event that triggered the replacement has already concluded, you have already met the "score" limit for that battlefield for the turn (Rule 465).

Furthermore, "Conquer" effects are triggered abilities that check for the state of the battlefield at the time of scoring/conquering (Rule 383.4.c). By replacing the battlefield after the score, the new battlefield (Trapping Grounds) was not the battlefield that was actually "conquered" for the purpose of the scoring event that triggered the replacement.

Sequence of Events

1. Combat/Showdown: You conquer the battlefield (currently Brush).
2. Scoring: The game triggers the score. You gain your point and trigger any "When you score" abilities.
3. Brush Ability: You use Brush's ability to replace it with Trapping Grounds.
4. Conclusion: Because the score has already finalized, and because the conquest happened while it was still Brush, the game does not re-evaluate the scoring event for the newly placed Trapping Grounds. Consequently, its "When you conquer here" ability does not trigger.

If you have the exact text for "Trapping Grounds," please share it! If it were a Passive or triggered ability that checks state differently, the interaction might change, but under standard timing rules, you cannot trigger a conquest ability after the conquest event has already finished.

This is my interpretation based on Rule 465 and Rule 466 regarding scoring limits. No official FAQ exists for this specific interaction.