If I capture 1 base to get to 7 points and then capture a base on my opponent's next turn, can that give me the 8th point for victory?

No, you cannot win the game by conquering a battlefield on your opponent's turn to reach 8 points.

According to Rule 466.1.b.2, when you attempt to gain the final (winning) point through a Conquer action, you must have scored on every battlefield through either method (Hold or Conquer) during that same turn.

Because you are performing the action on your opponent's turn, you cannot satisfy the requirement of scoring on all battlefields within that same turn. As a result, when you conquer that second battlefield on your opponent's turn, you will draw a card instead of gaining the 8th point.

This is supported by RiftJudge FAQ #9058, which confirms that conquering a battlefield on an opponent's turn cannot be used to reach the 8th point and win the game.