Is it possible to generate electricity through the earth's rotation?
Probably not in the way you may be thinking of electricity generation.
The earth already has a very powerful magnetic field interacting with cosmic radiation to produce some spectacular electromagnetic effects! However, harnessing those natural processes OR trying to use the earth as a magnetic core of a generator is not very likely.
An electric generator requires a moving part (rotor) and a fixed part (stator), one of which must be a magnetic material and the other must be an electrical circuit. Either way, building a generator based on the earth's rotation implies that we need to build the stator part outside of the earth.
Even if that were technically possible, there are SO many more efficient and available source of electric power generation that it would really not be worth developing.