Can I keep my opponent's battlefield a Brush if I conquer it?

Yes, you can. When you conquer a Brush token battlefield, you become its controller. The Brush's ability says "When you score here, you may replace this with the battlefield it replaced" — and "you" refers to the battlefield's controller (Rule 187.6.c). Since you now control it, the choice is yours. You may simply choose not to replace it, and the Brush token stays in play under your control.

The Brush token doesn't automatically revert or expire. It stays until its controller chooses to swap it back upon scoring, or until a game effect removes it (FAQ #8747). Conquering it doesn't force a swap — that's entirely your decision as the new controller.