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The Quiet One is for characters that speak little (as in, they will speak when necessary, but usually short and to the point). It is not necessarily correlated with being stereotypically cool.
There is Motor Mouth for a talkative character in general, and You Talk Too Much! for when someone voices their annoyance about it. Again, it does not necessarily mean they are lame or have specific reasons for it — it's a character trait. And although it is in the Character Flaw Index, sometimes even flaws can be put to beneficial use.
Edited by KD
I feel like this is tropeworthy.
"It's just a show; I should really just relax"
Is there a trope for a character, established as stereotypically "cool", talking very little with others - just saying 'sup', 'cool' and 'sure'?
The opposite, in that case, is a talkative character being lame, and talking a lot because he wants attention, is trying to impress people, or is unable to get social hints.