They are using rock, in the "rock n' roll" definition. I assume this is "How I Met Your Mother." When you ring something, like ring a doorbell, its loud but not explosive. When you rock something, people usually mean obscene levels of noise and celebration aka rock n' roll. Nay is a way to say "no, instead." I think the "ring" is a play on words or joke. This is because when you greet somebody, its usually because you "ring" their doorbell. So why ring their doorbell, when you can throw a rock n' roll party?

We say "Ring in the New Year" as a way to say we celebrate the transition from the old year to the new. It originates from the ringing of bells in many towns as a form of celebration. To say "ring, nay rock" is to imply that we will celebrate even more. "Rock" comes from "rock and roll" music, thus to rock in the New Year is to really celebrate.