Hi Bobbie...I'm glad that you took this poor little Green Cheek in, because it sounds like she hasn't had the best life up to this point unfortunately. If she was not hand-raised by her breeder but rather parent-raised by the mother and father birds, then you actually have to "hand-tame" her yourself...It's not a matter of her "realizing that you're a good person", it's simply that she was raised/fed by her bird-parents completely as a baby, and was not exposed to any people during that time period. So when a bird is parent-raised instead of hand-raised by a person, then they are naturally afraid of all people, because they don't know what they are or if they are going to hurt them, etc. And it also sounds like her first owner didn't make any effort to try to hand-tame her or even give her time out of her cage at all, so that's what you're going to need to do...
Now you've only had her in your home for less than a week, so right now everything is totally new to her, you, your home, everything. So it's going to take a lot of time for her to simply feel comfortable in your home and your presence. I would not let her out with your Sun Conure at all, as they may hurt each other badly. Right now you need to simply work on you bonding with her and you earning her trust, so that you can hand-tame her. Even if she was a hand-raised, tame bird just like your Sun Conure is, that still doesn't mean that she would ever get along with your Sun Conure. You never have any way of knowing whether any two birds are going to like each other, get along, bond with each other, or whether they are going to simply hate each other and be violent and aggressive towards one another. So you always have to be prepared to keep any new bird you bring home completely separated from any birds you already have. The other issue you may find that you have is that your Sun Conure, since he/she was in your home first and is already bonded with you, may have jealousy towards the new bird coming into your home and interacting with you, and he/she may become very protective and defensive over you.
So as far as "introducing" your Sun Conure to your new Green Cheek, the best thing you can do is to put both their cages in the same room so that they can see each other and talk to each other, and let them get to know each other that way for a good, long while. However, any time you bring home a new bird and you already have a bird,
you should ALWAYS quarantine the new bird in an entirely different room from the healthy bird you already have, for a minimum of 30-days! They should not be in the same room with one another for at least 30-days, especially with a bird such as your new Green Cheek, where there is evidence of feather issues and bad care in their prior home, as there are many Avian Viral Diseases that can cause feathers such as you describe, and there are also a lot of very common infections, both Bacterial and Fungal,
that birds can contract through bad care, such as not having their cages cleaned at least once a week, their food and water changed daily, etc. The first 30-days that you have a new bird you need to watch them to make sure they are eating and drinking normally, they aren't sleeping all the time, etc. All birds possess an innate, natural survival-instinct that allows them to hide any and all outward signs and symptoms of illness and pain. It keeps them from appearing sickly and weak to predators. So by the time birds are actually showing signs/symptoms of illness, they have usually been sick for weeks to months. This is why you have to do a minimum of a 30-day quarantine with any new bird you bring home, because even though they might not be acting sick that doesn't mean anything at all since they hide all outward signs of illness for as long as they can...So you should still remove the new bird and put her cage in an entirely different room from your Sun Conure for the next 30-days, even though they've already been exposed to each other, you may still be able to prevent any infections, diseases, or parasites that this new bird may have from being contracted by your Sun Conure.
Also, how large is the cage that the Green Cheek is in? The reason I ask is because a lot of the time when their tail-feathers are all broken and roughed-up, along with their wing-feathers, it's because the cage that they're in is far too small for them and their tails and wings are constantly rubbing up against the cage bars. Green Cheeks need very large cages, just as large as your Sun conure needs, because they are so active and playful. They love to climb, jump, swing, etc., and if their cage is too small then this can cause both physical and psychological health issues.
. It will be worth the time in the long run. You’re on the right track though - getting her used to being around you, offering treats, etc. Trying to ensure that your interactions with her are positive is definitely moving on the right track.
