Underground,
From your post, I'm assuming you're a middle infielder? Part of the process of the turn at second(like Vina) is proper footwork along with what you are asking about, a quick transfer.
Since you are looking for drills to help you with those "quick hands", I'll offer you one that I use with my players that has been very effective.
It must be done how Redbird describes, catching the ball in the palm and with an open glove(don't close it on the ball, just stop it). It's also easier to do with a smaller glove(say an 11 1/4" or 11 1/2" size) versus a large deep-pocket glove like an outfielders.
I call my drill "fingers up" and if I am teaching beginners, it is a three-step process.
First some rules. Must be in a balanced, athletic stance with feet approx. shoulder width apart. The fingers must point upwards(elbows pointing down) and both hands "out front" away from the body. The natural way for most people to catch a ball is with the elbows pointed out sideways(approx. a 45 degree angle), but with the fingers up/elbows down, I think with some practice, you will find that the transfer from the glove hand to throwing hand will be quicker and easier.
I start beginners out with no glove, just bare hands. After getting them in the described position, I will softly toss a ball to their glove hand and tell them not to catch it, but to just stop it and let it fall to the ground(don't close your fingers around it to catch it). This will demonstrate how the ball should be stopped(not caught) and where it should be done, in the palm.
Next, with their throwing hand beside their glove hand and still following the "rules", I will toss it to them again, but this time they must transfer it to their throwing hand and "show me the ball" as quickly as possible(don't let the glove hand waver from the fingers up position and don't close the fingers or begin cradling the ball near the body: keep the hands out front).
Once they have mastered that(which doesn't usually take too long), I move them on to using a flat glove-shaped training device(like the foam Soft-Hands trainer or have even used a thin piece of plywood shaped like a glove that can be strapped to the glove hand). There's no way they can close their fingers on the ball now and it forces then to learn to "stop the ball and transfer" without cheating in any way(still always fingers up and hands away).
After many repetitions, the player will eventually get the hang of this and can perform this skill faster and faster. Also, as they progress, make the throws from a little further away and with a little more on it.
Once they start using the glove for this drill, it will be necessary to constantly emphasize what has been learned(proper stance, fingers up, hands away, use the palm and not the web, don't close the glove), because they will want to revert back to their old ways.
As Redbird also stated, it will take many repetitions to effectively master this basic skill, but once it has, then it's on to the footwork phase(equally as important).
I know this was pretty long, but I hope it made sense. Good Luck!!!