I had idea that if the Holy Qur'an falls from someone's hand or he disgraces it, then he should pay some expiation (money or gold). Is this a correct understanding? Please explain and provide evidence from Qur'an and Sunnah.

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.

If the Quran is dropped from a person unintentionally, then he is neither obliged to give any charity nor to pay any expiation since there is no evidence for that. 

In fact, no act of worship is to be performed without an authentic Islamic evidence and there is no evidence about the above case.

However, voluntarily humiliating the Quran is apostasy from Islam. The person who has done so is obliged to sincerely repent and renew his Islam by pronouncing The Two Testimonies of Faith.

Giving charity or doing any other act is not sufficient as expiation.

Allah Knows best.