Thank you for your question.
It is possible, but it depends first on whether the inmate meets the criteria for a "Good Adjustment Transfer" to request that they be transfered.
Even then, the final decision is with the Florida Department of Corrections and is based on the needs of the FDOC and meeting the needs of inmates - security, health, etc. Getting an inmate closer to home, even if they are ill, so that they can visit with family, is not (if I am being honest) their priority or concern.
A transfer can also take a long time - a year or more would not be unheard of, even if approved, as space would have to be available at another facility that could accomodate your son. Here
is a link from the Florida Department of Corrections which talks about inmate transfers. If you scroll to the bottom, there is information about Good Adjustment Transfer qualifications, and information on that page as well as to how an inmate applies for a transfer.