I just got awarded a bounty in this site, but I don't see any hat appearing.
Is there any?
There was a Bounty Hunter hat back in Winter Bash 2013, awarded for winning a bounty:
Unlike other hats, this one wasn't used again, I guess because it caused many people to award bounties just for sake of the hat, and not based on the actual contents.
Unlikely. None of the regular hats have this trigger; it could be one of the secret hats, but most of those get awarded within an hour.
In the past, there have been hats for posting a bounty and also missing a bounty (e.g. This Is Fine from 2016) as a sort of consolation prize, but AFAIK never for being awarded a bounty (edit: see Shadow Wizard's answer, there was one back in 2013). The bonus reputation is enough of a reward already, I guess :)