Quadcopter drones rely a lot on the quick response of the four motors, continuously making small precise adjustments to keep the platform in balance. It would be quite hard to make a small turbine engine that has the same level of responsiveness.
However, there’s a huge problem of endurance. Batteries still have a low energy density per unit weight. Maybe it could be feasible to run a single onboard turbine to generate electricity and drive the propellers as usual.