Once we have configured the ABAP-Stack of a dual-stack system into the transport domain through CTS, so can we now move the JAVA-based objects as well or we need to do something else also.
Your question is pretty vague. As per it, we don't know if you have actually carried out all the steps of CTS+ configuration prior to including the abap stack in transport routes or not.
If you have already completed CTS+ configuration and this was the last step you did, then your java developers should be able to include their objects in a transport request.
However, if you haven't done the CTS+ configuration yet, then please do that. For reference, these are some documents that should help you:
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