I almost got a 2 year old green cheek conure for that very reason! Well, the daughter bought the bird for her mother, but the green cheek had already bonded to the daughter, so she couldn't be held by anyone else without her biting! And she even bit the owner on the lip! Daughter moved away to college, left birdy at home, and she is then in a family that no one can handle her! They were afraid to!
I went over to their house, fully expecting that I'd have to take this little green cheek home with me! Of course, deep down, I was hoping not to! I observed the behavior between owner and bird. The owner couldn't get the little GCC out of the cage, so I tried a different approach. Took a little longer, but it worked! I used sunflower seeds to motivate this little green cheek conure to come out of her cage and onto my hand! I had never met her before, so I didn't know how she would react to a stranger, but she did well! Once I got her out of the cage, I showed the owner how to work with her using food as a way to motivate her! Wiston (GCC) is highly food motivated so I was able to get her to turn around on my hand for a reward. The owner had problems getting Wiston off her shoulder, so I purposely put this 'strange bird' onto my shoulder, and I successfully got her to step down without a bite or a nip!
She did give me a nice good bite on my hand, but this had nothing to do with me being a stranger or something startling her, but had everything to do with the fact that I was restraining her with my left hand as I was trying to unhook her overly large leg band from one of her toes! If not for that incident, I doubt I would have ever been bitten!
I gave Wiston's owner new hope! And with a different approach to training, they have formed a stronger bond! It went well for several weeks and she did end up getting a bite (actually, that's when she got bit on the lip), but she stuck it through! And now the two are closer than ever before! Bites and nipping have been taken to a minimum.
The worst bite I've ever received was from Noel, my cherry headed conure (RIP). He bit me right in the webbing between my thumb and forefinger. It hurt, and it hurt A LOT! I hate to admit it, but the first thing that rushed through my head was that I wanted to bite her head off. Obviously, I didn't, but I felt horrible afterwards, for having that thought! I never thought about rehoming him after that, because I knew that he had no trust in humans. He bit me because I pushed him too hard in trying to earn his trust. In time, we did come to an understanding of each other and I learned when to expect bites, and I learned why they were happening. He was a special case though, because if I tried to teach him not to bite, I'd end up getting bitten more! He had stability issues, and he learned to bite to ensure that what he was stepping on was a stable perch. He only bit when he was unstable (usually because he tripped over himself, causing the instability). For normal, healthy birds, my recommendation is to avoid getting bitten!
Easier said than done, I know, but the only bite that can't be rewarded is the one that never occurs! If you haven't already, I would highly suggest clicker training, starting with target training. With a target, you can teach a lot of behaviors for your conure to do that are hands off, and if you give him direction and things to do, he may be less likely to bite. Also, through clicker training, you can learn how to better understand your bird and learn how to better communicate with him!
The media & blogs by Barbara Heidenreich, Lara Joseph, Susan Friedman and Melinda Johnson can all be great materials to help you!