I would agree with Jordan. I have never heard the phrase. I also can't find it online. My guess is that someone has made it up. I would agree with Merlyn that being new to teaching is the obvious interpretation, because of the connection between 'green' and inexperience.

I've heard of rednecks (people who generally work outside in the sun and are thought of as having little education or refinement). I've heard of people being described as green when they have little experience in a job. I have never ever encountered the term green-neck.

It means : " I'm not new to teaching."