No, Luffy got his first bounty for attacking the Marine that was working with Arlong, while Zoro got his bounty from beating a shichibukai. Beating a pirate doesn't get you bounty, as far as I know or remember, unless the pirate in question is somehow associated with the Marines or WG. Beating a shichibukai, just as Zoro did, gets you bounty or higher bounty because you essentially took on a part of the WG/Marines and beat them, which makes the WG look bad. A Marine can also create a bounty or maybe even raise it, just as that one Marine did to Luffy out of revenge for Luffy punching him for working with Arlong and stealing Nami's berries.
While it's debatable whether beating a Yonkou would increase bounty, it likely would if the victor is a pirate, because it means he could be another emperor and be more dangerous. The WG would likely fear the pirate that took out one of the strongest in the world.