I think you need to be extremely careful when making RPGs in someone else's copyrighted universe. In this case, I believe Paramount owns the copyrights to Star Trek. This goes doubly so when you use specific characters from that universe like Q. They might be fine with fanfics, but an RPG using specific copyrighted characters is much more risky from a legal perspective.
If I wanted to create a game set in the Star Trek universe, I'd create a very different futuristic sci-fi universe, with its own backstory and characters. And nothing says I couldn't create a Powerful Master Alien Race which places humanity on trial if I wished.
It's one thing to make something loosely inspired by another work of fiction. It's completely different to directly set a game in someone else's universe.
If the game is strictly for your own and maybe your friend's use, it might not be visible enough to be an issue. But, if you plan even to freely distribute the game, I would create a custom game universe. And certainly if you plan to go commercial with the game, you really need to talk to a lawyer and be sure you're legally in the clear. Most likely the latter would require some type of licensing deal with whoever owns the copyright.
That's probably why you're not seeing any chibi Star Trek type characters. It would be a legal minefield and nothing is more painful than a legal mess. Or more frustrating than working very hard on creating a great game and then having to rewrite the entire thing when you get a takedown notice.
