Can a female shake hands with her father in-law (her husband's father) when she greets him?

Praise be to Allah, the Lord of the World; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

It is permissible for a woman to shake hands with her father-in-law. There is no harm in showing him her external adornment that is usually uncovered in her house, such as her hair, neck, hands, feet and the like. Allah Says (interpretation of meaning): {…and not to reveal their adornment except to their husbands, their fathers, their husband's fathers, …} [24:31].
Know that father-in-law is among the permanent Mahaarim (unmarriageable persons) for a son's wife. By establishing a marriage contract with a man, the wife then becomes forbidden forever for her father-in-law (her husband's natural father).
Allah knows best.