It can be done, but not that easily and not that popular of a move either with the firms involved or with practicality.
You would have to either have the put moved to the firm that is holding you stock or have the stock moved to the broker that has the put.
All options positions are held by OCC, the holder of the put would have to have option moved the current brokerage firm to the new brokerage firm.
It would be easier to move the stock at the present broker to the broker holding the put. This would have to be done through the ACATS system which would be initiated at your request by the broker hold the stock. The broker holding the stock can, and probably will, charge you for moving the stock.
It may be easier and cheaper to close out the option and buy it back at the other firm or easier yet, just carry two long positions with the two brokerage firms
from the street