Well I have owned two Green Cheek Conures and I currently have a Yellow-Sided Green Cheek Conure and a Blue Quaker Parrot, both males....
They were both hand-raised, hand-fed babies that I brought home from their breeders right after they finished fully weaning, the Quaker at 8 weeks and the green cheek at 9 weeks. Now as already mentioned by everyone that has posted, there is absolutely no way to guarantee that any bird you get will fit the typical description of the species. Each bird is an individual, just like people are, and though we can all make generalizations about a particular species' personality traits, often a bird doesn't fit those generalizations at all. That being said, there are things you can do to enhance your chances of getting a "snuggly" bird. I would highly suggest that if your bird being a "snuggly", affectionate, loving bird that will want to be touched, will like being petted and like to get scratches, will snuggle up and nap on your shoulder, give kisses, etc. that you get a baby that has been hand-raised and hand-fed by their breeder (assuming you are talking about getting a young baby bird and you are not planning on adopting an adult bird from a rescue or one that is being rehomed). Any baby bird you get is going to go through an adjustment period the first few days or weeks, but just from my own personal experiences both from owning birds and breeding birds, if the babies are hand-raised and hand-fed by their breeders they usually tend to be very affectionate right from the start. Again, there are exceptions to this.
Quaker parrots tend to be a bit larger than Green Cheek Conures, as Green Cheeks are some of the smallest Conures. That being said, they are both in the same general size category, and it's not uncommon for a Quaker to turn out very small and Green Cheek to be unusually large. They are both going to be much larger than a budgie regardless, and there is absolutely no guarantee that any two birds will get along. Neither a Quaker or a Green Cheek will get along better with a budgie or any other bird, it's all about each individual bird and the individual budgie. I bred budgies for many years and I currently have 7 budgies that I bred and hand-raised myself that are every bit the pets to me that my larger birds are. The Quaker and the Green Cheek each have their own individual cages in my living room, and my 7 budgies live in a very large flight cage also in my living room, along with a Senegal parrot in his cage, and a cockatiel in her cage.
The bottom line to birds living together and getting along, no matter their size or species, is this: There is no way to predict if two birds will get along, and even if you introduce them to each other and they do seem to get along, they should never ever be left alone together without you supervising them. Your budgie is going to be much smaller than either of the species you're looking at, and it would take only a split second of them being together for the Quaker or the Green Cheek to severely injure your budgie, possibly fatally injure your budgie. I let my budgies out of their flight cage every day for at least 2 hours, but the 4 larger birds either go into their cages or I will take them upstairs and supervise the 4 of them in the spare bedroom that I have their big play gym in while the budgies are free downstairs. This is because the budgies are brothers and sisters and have lived together since birth and still do live together, but I do not ever allow any of the other 4 larger birds out together unless I'm with them, not even my Quaker and my Green Cheek, and you can see by the photo that they typically get along very well, to the point of them being bonded. But fights still happen, and a few months ago my Green Cheek (this happened in literally less than a second with me sitting next to them) managed to crack my Quaker's beak. So the bottom line is that you will never be able to allow your budgie to be out alone with whatever other bird you decide to get, and honestly even if you're supervising them out together they may never like each other to the point that they will always have to be separated and only get out of cage time separately. None of my 4 larger birds get along with any of my budgies, though they've all been slowly introduced and have been sitting next to each other in my living room, seeing and listening to each other for over a year now. As soon as any of the 4 larger birds is near any of the budgies out of cage, each of the larger birds immediately tries to bite the budgie, lunging at them almost instantly. It's not for lack of trying by me, but it just isn't going to happen. So we all have our own free time out in the house together separately.
As far as answering your questions about talking and being snuggly/bonding, again this is going to differ between each individual bird. There is no way to guarantee any of this, so keep that in mind. As far as talking, neither species are known to be excellent talkers, not like the very large parrots such as African Grays, Macaws, Cockatoos, etc. Both of my birds do talk, the Quaker by far speaks much more clearly than the Green Cheek, and the Quaker seems to pick up on words and phrases much more easily than the Green Cheek. The Green Cheek has a very rough, gravelly voice, while the Quaker sounds a lot like me when speaking. So as far as my specific birds the Quaker is a far better talker, but I'm sure there are exceptions to this as well.
The Green Cheek Conure is known as a "Velcro Bird", meaning they always want to be with you, on you, stuck to you...And that has been completely true with both of my Green Cheeks. But my Quaker is very affectionate, snuggly, and always wants to be with me too, so I can't really say that one species is more affectionate than the other. They are both bonded very closely with me, but I truly believe that this has to do with how you interact with your specific bird, how much time you spend with them, how you treat them, how you talk to them, how often you talk to them, how often you take them places with you, whether or not they are located in the main room of your house or not, how often you expose them to other people, etc. I am single, I live alone. And my birds are my family, as well as my 2 dogs and my bearded dragon. All of my bird's main cages are located in my living room so they are always where the action is. They always get to interact with my friends and family that come over. If I'm at home they are all out of their cages (except for the 7 budgies that get 2+ hours at least by themselves every day). Most often if I'm home I'm in my living room and my 4 larger birds are sitting on either their cage tops or their individual play stands there with me. They play with each other or by themselves, they nap, they sit and talk, I'm not always interacting with them, but they are out and with me, just like young children would be. The 4 larger birds each have their own small sleeping cages in my bedroom upstairs, and at their bedtime I take them up and they go to bed, I join them later. The budgies sleep in their big flight cage in the living room, and often I let the budgies out after the larger birds go to sleep. If I'm in my office upstairs working on the computer or I'm in my shop restoring or repairing a guitar, all 4 birds are in the room with me, usually on their play stands. They go into their cages in the living room whenever I leave the house, but as soon as I'm home they're back out.
So you can see that my birds are all very affectionate, snuggly, and very bonded to me, and I believe that this is a combination of both them being hand-raised babies and them living as a part of my family and always being with me. They also go on their Aviator Harnesses out shopping with me, for car rides, for walks with the dogs, etc. So if you want a snuggly, affectionate bird, either a Quaker or a green cheek will fit the bill, but only if you raise them as such. Now the fact that you live with other people in your home can definitely change things, because often birds do become attached to one particular person more than the others in the house. If you want to be that person in your house you need to be the one to always be with them, to feed them, give them treats, play with them, nap with them, etc. If you were to pass off their basic care to someone else in the house (as kids often do to their parents) then it's possible the bird will choose that person who is feeding them and spending the most time with them as the person they bond with. They may like everyone in your house but you're their person, or they might hate everyone else, it's all in how they are socialized in your house, which is why I suggest keeping their cage in your living room where everyone in your house spends time, and where other visitors will interact with them.
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