Curly coming to silly conclusions about her own name wouldn't explain why she comes up with such a specific name for Quote.
I prefer to think that they both had a level of sentience and formed a strong friendship prior to the memory loss. Based on this, one or the other probably wrote her name on his hat.
Perhaps Quote lost the ability to read for a brief while. He clearly has the ability when he begins reading signs though, so that's out the window. Maybe he just never looks at his hat? Maybe his hat doesn't come off and he never looks in a mirror?
The major hole in the game though: it's pretty clear that other characters can read too, since they leave notes for Q, or create the town signs. So why wouldn't they read Q's hat and ask "what is Curly Brace?" The only thing I could think is that this game is in multiple languages. Q is bilingual and can understand the humans and Mimigas, but they can't read the writing of the language of his hat.
This would of course, clash with pet theories that Q/Curly were designed by the expedition team. The problem there though is that the team has Japanese names and presumably reads it and the kanji is Japanese.
Another problem is that Curly knows her own name, yet doesn't seem to react to seeing it written on Q's cap.
Then again, Curly doesn't react to you having her panties either... so... either Curly is VERY absent-minded, like she's afraid to look at Quote closely, or she's keeping a lot bottled up.