Can I change my student visa to a work visa without leaving Thailand?

If you're on the original 90 day non-ED visa then you have no options. The only thing you can do from that is apply for an education extension.

If you're on an extension of stay, where the stamp in your passport says "extension of stay" then maybe. Most likely you cannot. You would need to check with your school, your employer, and the immigration office to see if they will coordinate to switch you from education extension to work extension. I'm not even sure that's possible since they are completely different visa categories. But the immigration office can tell you if it's possible.