Green Island is a Metaphor for School
It said on The Other Wiki that Quack is intended to be a few years older then Peep and Chirp, (immaturity aside) who might qualilfy as kindergarteners and first graders, who are old enough to go but not old enough for homework, essays, long division, letter grades, etc. Quack is probably fourth/fifth (or even third, depending on how strict this particular school is) grade and is probably just beginning to wet his feet (no pun intended) with those sorts of things. Quack, pompous little nipper he is, probably gets pretty high-and-mighty talking about how smart and mature he is, so much so that the naive Peep and usually less easily impressed Chirp are both awestruck and can't wait to go there. And then in one episode, Quack actually does take them there, they find it to be not quite as exciting as they thought...something that many young ones can relate to. Quack used to be a kid's pet
Which would explain where he got his hat from. Said kid probably was a huge fan of ducks, which also explains a lot about Quack. Since Quack seems to be pretty fond of the hat, it's likely the kid treated him well, but at some point let Quack go (naively thinking that Quack would be able to survive on his own-luckily, apart from Tom, their area seems pretty safe) He probably got his weird ideas (like his version of how the big wide world came to be the way it was, and his trippy dream in Peep's Moon Mission) from kid's books that the the child read to him. Pink Quack is possibly a replacement goldfish the parents got for the kid...but apparently they didn't bother to teach the kid not to do the exact same thing he/she did before. Robin from "Big Bird" and "Chirp Flies the Coop" is Chirp's Future self.
She has the same three black hairs and is surprisingly chill about this strange little girl following her around. (And how easily she deals with Quack) It also adds new meaning to when she tells Chirp that she used to be just like her. It also makes more sense that she knew all along Chirp wasn't able to fly and was only pretending to not realize for the sake of keeping a stable time loop. How did grown-up Chirp get sent back in time? A few ways: Either magic, time lords, or Peep started discovering worm holes and quantum theory after he out grew caterpillers and sunflower seeds. After she got sent back, she realized what had happened from her memories of when she was younger, and did her best to replicate what had happened based on those memories.- Or maybe she's Chirp's mother.
Quack's voice sounds different In-Universe.
Quack is clearly a duckling, with the appearance, maturity, and knowledge of a juvenile. But he has the voice of an adult-likely because Quack is the comic relief, and it would be hard for a child to be able to deliver Quack's rhymes with the comedic timing the role demands. So Quack is played by an adult. Like how Tom sounds like a woman even though he's canonically a male kitten. In universe, Quack's voice lacks Larynx Dissonance. Quack goes to Green Island to be alone.
Basically, Green Island is Quack's place to be alone and sleep with no distractions from any of the other animals and his friends. Which explains why he hypes it up so much and why Peep and Chirp find it boring when they go there.