connecting devices with two different power supplies
In order to turn on Q1, there must be a base current.

If the grounds are actually electrically isolated, if there is no electrical connection between the two grounds, there is no closed loop for base current through the 3.3V source.  You won't be able to turn on Q1 with the micro-controller.

Either the two grounds must be connected together somewhere or you'll need to use, e.g., an optocoupler to drive the base while maintaining electrical isolation between the grounds.