The bounty quest is a special quest obtained from the arms dealer the rewards are mostly random as it is usually just money and exp by percentage. However sometimes you may get a weapon from these quests if you take a look in the quest screen and hover over the reward boxes you can determine if it's worth going after that one after accepting it which then you can delete the quest and move on. It is not important for progress to perform these quests but may be great early game some might help even in hardmode
If anyone wants to put the sprites and stats for bounty-quest related weapons and items here feel free
Trivia[]
Nakano has implemented a Bounty Quest system like this on his TerraGuardians mod, where Sardine can offer you Bounty Quests from time to time, after you speak to him about them, and place a sign or announcement box near the place he stays on his house.