Accepted in TurboTax just means that they have received your return but have not processed it yet. This is normal; until they start processing it on their own, you won't find record of it when tracking on their website. So if you contact the state or try to look up your refund status, don't be surprised if they tell you "no record found." Most states have some type of note on their refund tracking page that states how long processing may take. 

The accepted status in TurboTax will remain the same from here on out, since we no longer deal with the return once the state or IRS receives them.