Zerodha is a DP (Depository Participant) under CDSL. So, the demat account would be held with CDSL as the depository and not NSDL.
I assume that you also hold an NRE-PIS account with Axis Bank. In this case, we can link the same Axis PIS account and open the Zerodha trading & demat account.
You would be required to close the Axis direct trading & demat account, You cannot hold multiple accounts across brokers/DP. You will have to first open the account at Zerodha and then initiate the closure at Axis direct.
You can ask Axis direct to do a closure cum transfer. Check this link to understand the process of transferring shares from Axis direct to Zerodha.
Please refer this link for the process of opening an NRI account. For more information, refer this article on Z-connect and this video on NRI account opening.