Build a simple coil generator
Firstly, current in not induced; voltage is induced. You get current because of the induced voltage in the receive coil is connected to a load (or a short circuit in some examples).


  Does the insertion of a ferromagnetic material inside the coil
  increase the output current of the generator?


The ferromagnetic core attracts lines of flux from the magnet and concentrates that flux through the receive coil (more so than if the coil were air-cored). That means you get a bigger induced voltage when the magnetic flux is changing.

This usually means more current produced when you load the coil. 

However, because the inductance of the receive coil has increased (due to the presense of a ferromagnetic core) and that component of inductance is in series with the induced voltage, the current produced into a load may not be any bigger when the magnetic field changes are very fast.