We know that conducting materials can be heated by electromagnetic induction. Is it possible to generate current using a cooling process?
Short answer is: If you heat up the whole material, then no.
If you heat up the material in on end (an iron bar e.g.), then yes. The thermoelectric effect will cause a small current to flow between the two ends of different temperature.
It looks like you are mistakenly assuming that electromagnetic induction causes heating, which should then be possible to reverse. Well, the heating is more a side effect. It is caused by the resistance in the material. Whenever current flows - regardless of the cause - resistance in the material will cause heating. Electrons making up the current e.g. will bumb into the material atoms and transfer their energy to this material. Which is heat on the macro scale. This heat loss is an irreversible process - you cannot "put the heat back" to regain a current.