No, being talkative isn't that important, but being social is important. There is a big difference.
My favorite TV series is the Sopranos, and Tony Soprano, a true alpha male, identifies himself as "the strong, silent type" and he gets all kinds of pussy in the show, and he is seriously overweight. While I understand this is a TV show, I believe it does hold conducive to reality.
Think about some of the greatest alpha males you know, do they chat your ear off? I really doubt it, talking too much takes a lot of effort for approval seeking. The alpha male says as little as possible to get a point across, he isn't short, but he clearly communicates. You just need to be able to hold your frame, and not be afraid to say what is on your mind. I honestly believe you can seduce a girl without saying anything at all, and I have gotten close a couple times
You must be social though if you are going to get girls, you shouldn't be in the corner by yourself. Alpha males have a supporting cast. Interact with them, but don't chat their ears off.
A lot of times, less is more when it comes to talking. I create so much more sexual tension by looking girls in the eyes, by my body language, and by touching them, than I do by talking.