As you are new to the world of parrots and in particular to Quakers I offer you two links to help you understand you little Chloe a little better.
http://www.birdsnways.com/wisdom/ww12eiv.htm and this one on body language
http://www.birdsnways.com/wisdom/ww15eii.htmThe first one is an entertaining introduction to the nature of the Quaker parrot and the body language one will help you to understand things that she tries to tell you during your interactions with her.
Since you tell me that she is left out of her cage during the day with no supervision I am assuming that you have no other animals such as a dog or cat in the house. I am concerned about you doing this even if their are no other animals as a human area is a very dangerous place to be if you are a little bird. What if there is an open container of water that your bird gets into and can't get out of, such as a glass of water or an open toilet. There are always those pesky electric cords just lying in wait for a curious bird to chew on. There are little nooks and crannies that must be investigated that may catch a bird and from which it can't get out of. There are fabric to get those cute little toenails caught in from which many a small bird could not escape from and either chewed off its foot, received other serious injuries from such as displaced hips or bones broken in the attempt to escape or even coming home to find your precious bird hanging dead from a curtain. Human areas are dangerous for an unsupervised bird and even more so for an unsupervised bird. I am not an alarmist and my birds live with me in human living spaces, but they are always supervised when out and about and I am always on hand to help them if they have a problem. Thankfully there have been very few of these in the few years that I have had bird friends in my home.
It is not uncommon for many people to confuse being afraid of hands with merely avoiding them due to our use of hands to move them to places that they may not want to go or to be removed from a place that they prefer to remain at. Many times this hand aggression is directly due to our removal of them from a favored perch to place the bird in its cage. They realize that we are asking them to step up on to our hand from our shoulder and then we put them in the cage, they prefer to be on our shoulder and try to let us know this at first by just not stepping up or by avoiding the hand and then finally when we don't listen to them by biting the hand. Bad Hand, Stay back. Chomp, chomp.
Try this. Don't wait until you are short of time to ask the bird to step up. Ask for step up, if the bird does give it a treat and take it to another perch and step down. Wait a few minutes, ask for step up, if bird does not, wait a couple minutes ask for step up, if bird steps up go to top of cag and step down and give treat. Wait s few minutes, ask for step up, when bird steps up place in cage, step down and give two treats close door. Vary this procedure with four or five different perch places and change the order so bird never knows when it will go to top of cage or in cage and don't always leave bird in cage when placed in it. This is likely to solve the issues that you have described.
sound to me an awful lot like begging to me although I can't be sure without seeing and hearing this behavior, a video would be helpful here. It sounds like she is clipped, as most birds of this age won't stay on one perch very long especially if there is a human to perch on nearby. So does this perch have food and water cups built in or not? She could be begging for food and water or she could be begging for companionship and if this is the case either or both should be given. If she is not hungry and you can't hold her or let her perch on your shoulder, you should at least answer her calls to you. A nice easy two note whistle would work very well for this and in time she will also use this whistle if she is calling for your attention. This is a flock call and they usually will call for you if you are not in sight or are too far away.
Hope this helps you.