The quillrats are one of the wild pets that have the potential to be color-changers. If it shows up as a different skin in the Pet Journal than the one you captured, that usually indicates it's a color-changer. Summon it out a few times and see if it changes colors each time you summon it. (It could take a few tries to actually see a change in colors.)
If you're looking for one skin and only one skin, some have experienced that if you capture a pet as the primary battler, the skin won't change. But if you capture it as a secondary, it has a higher chance of being a color-changer. This isn't always the case, but many have reported that it's happened quite often.
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