Yes, bounties are handed in via the Contacts menu (Combat bonds, which is something similar, only from conflict zones, too)
If you have a general superpower-wide bounty to turn in (for example in Federation) then you can hand it in wherever you want in Federation space.
However if it is a bounty tied to a specific faction, you will only be offered to turn it in in a system where said faction is present. If you check your left panel - the Transactions tab, it should tell you. Most of the factions have a system name in their own name. Then you know you have to go to that system. (It's usually something like "Laedla Blue Mafia" or something, so you know for example that you have to go to Laedla)
If a faction doesn't have a system name, it's a bit unfortunate. It doesn't happen often, but I've encountered these. In that case you have to search for said faction in one of the third-party tools. (
https://eddb.io/system and
http://inara.cz/galaxy-minorfactions have the most up-to-date databases, as far as I know)
That's for the "normal bounties", however if it is for the mission (you see the orange globe mission icon on the system in galaxy map and next to the station in system map), check the Mission Board. If it is something to hand over to a faction in that system, one faction will have a small circle with an exclamation mark, next to their name. Click on it and you should be presented with an opportunity to finish the mission.