Lenz’s law states that the polarity of induced emf is such that it tends to produce a current which opposes the change in magnetic flux that produces it.Yes, emf will be induced in the rod as there is change in magnetic flux.When a metallic rod held horizontally along east-west direction, is allowed to fall freely under gravity i.e. fall from north to south , the intensity of magnetic lines of earth’s magnetic field changes through it i.e. the magnetic flux changes and hence the induced emf in it.