Can I extend my ED visa for my internship in Thailand by 60 days without leaving the country?

I have never seen an ED visa issued for internship get extended EVER <- as in not a single time.

While I personally believe it is a fruitless endeavor, the people you need to be talking with is the place you intern at (to see if they could supply you the documentation to get the extension AND the immigration office in the area where you intern to see if they would allow you to apply).

You have to exit, re-enter the country on a visa exempt OR go to a thai consulate nearby with documentation to get a NEW Non-ED visa based on internship. That's the only way I have seen people do it