Yes, you can transfer transfer the person with a Visit Visa onto your Work Visa but for that she will have to exit Qatar, and only then can you start procedures for the transfer.

A Free Visa is the same as a 'Visa with a NOC' available from the sponsor. That means she can work for you without the change in sponsorship as well. If you want you can transfer her onto your visa, but thee are certain guidelines for that of which am not sure of.