Welcome to the community and congrats on your little Green Cheek! They're awesome, loving little birds and just wonderful companions and family members...
As already mentioned above, 4 days home is like 5 seconds in "new parrot time", lol. If your new little baby Green Cheek is regularly stepping-up for you and interacting with you and your hands with no issues like you describe, then he/she was no doubt hand-raised by his/her breeder, and it sounds like you got a loving little baby...But as already mentioned above, he/she has only been in your home for 4 days...So she went from being with her breeder, who hand-raised/fed her, and her clutchmates/siblings, to being alone in a new home/place with new people and no siblings around. So as you actually mentioned, in the first 4 days of her/him being with you, she has established her cage as her "Territory" or her "Safe-Space", and it's the only Territory/Safe-Space he/she has right now in your home, so that's where she feels comfortable right now. In addition, in only 4 days there isn't anyone in your home that has earned his/her trust yet, which is the very first step to building a relationship/bond with a new bird, so she also doesn't have anyone that she has chosen as being "her person" yet, so she's not yet comfortable with one person enough to hang-out on/with them either. So right now her cage is where she feels the most comfortable, and that's why she's constantly staying on it/in it/around it...
I'm assuming by the way you described it that you have your bird's cage located in the "main-room" of your home, meaning the room where you guys spend most of your time whenever you're home (usually the living room, family room, etc.), which is exactly where it needs to be, so that he/she is always "among her Flock" whenever any of you are at home. Just being in the same room as him/her as much as possible will gradually make him/her more and more comfortable with all of you, and she'll slowly start to feel more and more comfortable away from her cage every day...
In the meantime, I would try to either buy or build her some other "territories" that are just hers, like some different floor and/or tabletop perches, playstands, T-stands, etc. that are located in different rooms around the house and in different places. This is going to allow her to feel more comfortable in different places all over the house so she'll start to stray away from her cage. It's really convenient and helpful to have at least one play-stand/T-stand that is portable and can easily be moved from room to room, so that she can be with you/someone in the house no matter what room you're in, and this will not only help to earn her trust and build a bond with her, but it will also help tremendously to start getting her feeling much more comfortable away from her cage. The stand doesn't have to be anything big or fancy, and you can make a portable T-stand out of PVC piping/fittings from Lowes and some Vet Wrap from Tractor Supply for under $10. Or you can buy a portable perch/stand at most pet shops...
The other thing you might want to start doing, if you're not already, is take advantage of the time period after dinner but before bedtime, as it's typically this time when birds are the most docile and cuddly, because they are tired and they want to be with their Flock snuggling. I don't know what you guys do at nighttime before bed, but if you are TV/Movie watchers, there's no better way to bond with your new bird than to shut the door to their cage so they can't go back in, turn the TV on and the lights out, and then curl-up on the couch or in a chair with a blanket and watch TV or a movie with your Green Cheek on your shoulder...They typically absolutely love this time of the day, and most Green Cheeks love nothing more than to cuddle-up with you and do some "beak-grinding" while on your shoulder, in the crook of your neck, behind your ear, on your chest, etc. while you give them some head/neck/chin scritches. This will help them to feel more and more comfortable with you and your home, strengthen your bond with them, earn their trust, and get them feeling comfortable away from their cage.
So the general idea is interacting as much as you can with your new bird AWAY from his/her cage, and to get them to start feeling more and more comfortable in places away from their cage, whether it be on different perches/stands or on/with you and your family...And something else that you need to remember that is extremely important is that you do still need to respect their "Safe-Space", as a lot of birds will stay extremely territorial about their cages for their entire lives, and even though they are the most loving, cuddly birds in the world, they still do no want your hands inside of their cage whenever they are also inside of their cage. I have an almost 4 year-old Quaker who I've had since she was 12 weeks-old and who is just a cuddlebug, but who it this day I have to open up her cage door, let her come out on her own, and THEN I can put her on my shoulder or on one of her stands or on top of her cage while I change her food/water and clean her cage out. If I open up her cage door and just stick my hand right in to get her out or to remove a dish, I will get an immediate warning-nip, and if that doesn't get my hand oui I'll get a good bite...But if she's outside of her cage then I can do whatever I want inside of her cage...So keep that in-mind for the future...
The bottom-line is that you're only 4-days in to a 30-40 year journey with your new baby Green Cheek. A month from now things will be totally different, and a year from now totally different than that...It just takes time, patience, and consistent daily interaction and love..
four days isn't much time, lots if getting used to each other and the environment.
