If your opponent pulls your unit to a battlefield but you end up winning the showdown, do you get a point for conquering?

Yes, you can score a point for conquering a battlefield on your opponent's turn if you win the showdown there, provided you haven't already scored that battlefield earlier in the same turn.

According to RiftJudge FAQ #3657 and FAQ #5834:

• You are eligible to score 1 point for conquering a battlefield per turn, regardless of whose turn it is.
• If your opponent pulls your unit to a battlefield (e.g., using Charm) and you win the resulting showdown, you establish control of that battlefield.
• If you have not already scored that specific battlefield during the current turn (either via Hold at the start of your turn or a previous Conquer), winning that showdown constitutes a Conquer action, and you will score a point.

Important Nuance: Per Rule 465, you can only score each battlefield once per turn. If you held that battlefield at the start of your turn (Scoring via Hold), you cannot score it again via Conquer later that turn, even if your opponent moves your unit away and you win it back.