Yes, there will still be induced current. Actually, the result of the experiment will be the same.
In fact, Faraday Neumann Lenz law says that a variation of the magnetic flux enclosed by a circuit produces an induced electromotive force in the circuit itself (and so an induced current), and the intensity of this e.m.f. is equal to the negative of the variation of the flux in time:
If we are moving the magnet towards the coil, we are increasing the magnetic flux through the area enclosed by the coil (because the intensity of the field becomes stronger). If instead we moves the coil towards the magnet, we are still increasing the magnetic flux through the area enclosed by the coil, because the coil is coming closer to the magnet and therefore the field intensity is increasing. Therefore, the net effect will remain the same, and so the induced current will be the same in the two situations.