If I move off of a battlefield leaving it open, and then acquire it again with a different unit on the same turn, do I still trigger Conquer effects even if I don't score a point?

No, you do not trigger conquer effects in that situation.

According to Rule 447, a player may only score from either method (Conquer or Hold) once per battlefield per turn. Furthermore, Rule 448.2.c states that Conquer abilities "will only trigger when the Battlefield is Scored; I.E. These cannot be triggered more than once per turn for a player."

As clarified in the FAQ:

• "You cannot trigger the conquering ability because conquering is a subset of scoring. If you can't score the battlefield (because you already hold it [or already scored it this turn]), you can't get conquer triggers."
• Even if you move a unit away and move a different unit back in, or if the battlefield becomes empty and you retake it, you cannot trigger conquer effects because you have already scored that battlefield once this turn.