If my enemy Bellows Breath my Rengar at a battlefield, does using Flash to change the location of Rengar cancel out Bellows Breath?

No — Flash does not cancel Bellows Breath.

Bellows Breath targets the units themselves, not the location they're at. "Same location" is a targeting restriction checked when the spell is cast, not on resolution. When Rengar moves to base via Flash, the spell tracks him there, and base is a valid location — so he still takes the 1 damage (FAQ #12002).

One nuance: if Bellows Breath also targeted other units that stayed at the battlefield, the controller picks one location among where the original targets now are and deals damage only to targets there — so Rengar could avoid damage only if the controller picks the battlefield instead. If Rengar is the sole target, he always satisfies "same location" with himself and gets hit no matter where he goes.