You only have to fight Kid Quick once. He is removed from the game after you earn the World Champion title.
Kid Quick is tough. He is not physically dangerous, especially when you get used to his timing. He can, of course, knock you out but the greater danger against him is running out of time. He does not provide too many opportunities to hit him, and he has no special move that you can hit him out of to knock him down.
Kid Quick tends to leave his body open more than his face. He will usually throw jabs in series of three. You can tell when he is about to jab because he raises his gloves, which allows you to hit him with a body blow. Alternatively, you can dodge his jab and try to hit him back. He sometimes follows the series with a hook.
The best opportunity to get a few hits on him is after his hook. He will raise his gloves in a specific way, different from the jab. Dodge it and throw a couple of left jabs. He will often let you get in a couple of extra jabs even after he is no longer stunned. Whatever you do, don't use body blows in this situation, because he will block and no longer be stunned. You can get away with making your first punch a body blow, but once he's stunned you must use jabs.
The breakthrough in a given match against Kid Quick will probably be when you finally get your KO meter full. He will often stand there and let you hit him multiple times. Don't stop unless he starts fighting back against them, just as Bald Bull does sometimes. If you get particularly lucky you can fill up your KO meter and knock him down very fast at the start of the fight, and he might not even get up!
You may remember that Bald Bull will occasionally block and react to a left jab in such a way that it opens him to attack. Kid Quick will not fall for that trick. He will, however, fall for the same thing with right jabs. Sometimes he will counter with a body blow to your right (dodging left is much easier than dodging right), but more often he will duck your jab and throw a hook. Now you can get in at least two jabs, or a jab and a hook.
Alternate method: This guy is actually the easiest person to beat in the game. Throw a right to his head and it will force him to retaliate with a hook. Dodge to the right and then follow with 3 left jabs to the face. Repeat this pattern and he goes down in about 30 seconds. 2 left jabs and a right to the face will force the hook retaliation, speeding up this process if Quick is not as full strength. This is touched on above, but with this method you are forcing him to throw a hook at you vs waiting for him to do it on his own.