A Crown of Stars: Daniel is the God-Emperor of Avalon, an Empire which spans a chunk of The Multiverse. He is the Big Good of his realm, powerful, ancient, and ever-concerned with protect his subjects and help people in need. He is also "quirky" to say the least. It's deliberately invoked, however, since interacting with people in a friendly and lighthearted manner helps him keep in touch with humanity; that, and anytime Shinji and Asuka see him he is usually "off duty", making culture pop jokes or wearing Hawaiian shirts while sunbathing.
Evangelion 303: Mari is nuts. She is an Elvis Presley fangirl, sings and makes "Pew, pew" noises during the battles and she will forget about an ongoing mission to try to beat a strong rival. And still she is good enough to give the base's top pilot a run for her money.
In Tiberium Wars, GDI Commander Karrde finds Lieutenant Kirce James to be pretentious, self-important, and overly dramatic, which annoys him to no end. At the same time, she's able to whip up squadrons of Firehawks, an Ion Cannon, an entire division of GDI Marines, and a thousand Mammoth Tanks for him on short notice, so he doesn't complain at all when it comes to the results.
Dr. Kaminko from The Pokémon They Carried has a habit of naming his military hardware after Pokémon. Command lets him keep doing this because he's just that damn good at designing weapons.
Between his incompetence at the digimon card game, lack of experience, inability to make effective plans and shy nature, it's easy to write off Takato from the Tamers Forever Series as weak excuse for a Tamer. but his outward awkwardness masks an astonishing degree of comptetance.
Fridge Brilliance time: It is actually the above weaknesses that make him so dangerous, his lack of long term thinking is made up for by his impressive skill at using surprisingly effective Indy Ploy's, making him an unpredictable and dangerous opponent
Sunny Breeze is absolutely this in Racer and the Geek, even if he's relatively less quirky than many of the examples here. Switches constantly between the three languages he speaks, knows more about firearms than the rest of the bank security outfit put together, drinks spirits like water, personally owns enough gear to outfit a fireteam, is afraid of talking to girls, and is a Shell-Shocked Veteran with an itchy trigger finger. He keeps his job for two reasons. Firstly, he is one of the few who can do the job. Secondly, His boss is his longtime friend. Thirdly, he is a very competent fighter with amazing instincts and a strong sense of honor. Fourthly, he hasn't crossed the line so such an extent as to demand something be done about him. In one chapter, they discuss that Sunny is likely one single Wham Episode away from being terminated, should anything happen that even being friends with the boss can't settle. Think of it this way: Sunny's idea of what to do if being a guard doesn't work out is to return to being a freelance mercenary! Being a BEL in this fic is not a good thing whatsoever.
The Ranma ½ fanfic Girl Days revealed that the Kuno family was full of Bunny Ears Lawyers, such as a WWII fighter pilot who wore a sailor fuku, or a samurai who fought with a live rabbit strapped to the top of his head. The current generation, though, are simply nuts.
Actually the current generation is pretty impressive. Tatewaki is a swordsman of such caliber and skill, he can use a bokken (bamboo training sword) and perform feats that surpass what is usually considered possible with a proper metal blade. Principal Kuno is a master of disguise, stealth, and using electric razors as weapons and has techniques that can cause weapons to split like banana peels. Kodachi is a mistress of poisons and rhythmic gymnastics combat. Despite delusions of grandeur, Tatewaki is actually considered quite handsome, and is very skilled in romantic gestures, just a bit misguided in who he directs them towards and oblivious to certain facts (mainly that female and male Ranma are the same person, despite Ranma transforming in front of him, both directions, on multiple occasions, including at least once while he was embracing female Ranma). Kodachi is completely psycho crazy, but she is actually pretty tactically skilled, and able to hold her own in fights with Akane, Shampoo, and Ukyo... sometimes simultaneously. The difference between the current generation and the previous ones is that the quirks of the previous generations never directly impacted anyone other than themselves.
ALL of the scientists in the Bleach fanfic Project Tatterdemalion fall under this trope: Urahara, Yoruichi, Tessai, Isshin, Masaki. The only reason they're still employed after all their insane prank wars, bizarre personal quirks, and failure to obey military protocol is that their scientific work is unimpeachable.
Shunsui does a much better job of keeping it under wraps but he's pretty eccentric for a high-ranking security officer.
Discussed in The Wrong Reflection. Prior to the fic Eleya avoided being cashiered for gross insubordination (cussing out Rear Admiral Tuvok and several dignitaries) because she helped save Earth Spacedock and Qo'noS from the Undine immediately after that. She kept her command and even got a medal, but she also ended up with a black mark on her service record that means she'll probably never make admiral (not that she wants to). The author explains the thought process here.
Ichigo in The World In Black And White. He's a bizarrely child-like, candy-obcessed Hollow, who is up to something involving notebooks, while still being every bit as overprotective and capable of kicking ass as his canon self.
In The Circle Xenophilius Lovegood is considered an excellent political tutor despite such personal quirks as sniffing the top of Harry's head and asking Neville if he likes pudding as much as Luna does.
In Sonic X: Dark Chaos, this trope is the pretty much the only reason Sonic and friends even let Eric anywhere near them. Despite being an Idiot SavantCloud Cuckoolander, he's got the technical skills of Tails with the fighting skills of Sonic - and he's genuinely good (most of the time) under all his insanity. This is also the reason why Sonya deeply loves him despite his craziness.
Cynthia in Secret Dreamer is just as eccentric as she is in Awakening, but she shows herself to be a capable commander during an unexpected skirmish with the Risen, deftly managing the recruits and striking more than a few decisive blows of her own.