BF1 is an open battlefield. I move Yone, Blademaster to BF1, then my opponent play a Rengar there. I win the combat and conquer. Can i use the yone ability, since its started as an open battlefield?

Yes — Yone's ability triggers.

The condition is "conquer a battlefield that was uncontrolled" (current errata text). BF1 was uncontrolled when Yone moved in, and it stayed uncontrolled the whole time: control can only be established at the end of a Showdown or Combat (Rule 190.4), so Rengar joining as a "surprise defender" never made the battlefield controlled. When Yone wins the combat and establishes control, he conquers a battlefield that was uncontrolled — the trigger fires, dealing 5 damage to an enemy unit in a base.

This matches the established "surprise defense" ruling: a defender arriving mid-showdown doesn't change the battlefield's uncontrolled status, so Yone still triggers on conquest. (Distinguish this from the case where the battlefield was already controlled/occupied when Yone arrived — that conquest would not trigger.)