The former is definitely the more common of the two. I'm not sure if 'treat' is specifically British English, or just old-fashioned, but either way, I would never use it. I would just say 'dinner', rather than 'a dinner'. I think that the way I would say it would be 'I would like to take you out to dinner'.

Treating someone to something means that you're giving it to them and you don't expect anything in return. Buying something for someone can be a treat, but isn't necessarily so. When you say "I can buy you dinner" you may expect to get repaid the cost of the dinner, although I would probably say "I can pay for your dinner" in this case (this is still a little ambiguous, would be better to say "I can lend you the money (for dinner)" ). If you did intend to treat someone, a more modern statement (though it may already be old-fashioned now :) ) would be "Dinner's on me".

Buy you dinner would be the phrase more commonly used...treat you to dinner is more formal.

"treat" means sharing