After hand shaking, somebody put their hand toward their chest and some people don't do this. Please kindly tell me which action is correct according to Qur'an and Sunnah?

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

What you mentioned is included in the habits and customs of people. Such habits are lawful unless otherwise stated in Islamic Shariah. So, there is no harm to put your hand on your chest after greeting somebody. However, not doing so fits more to the Sunnah, as this act - to our knowledge - was not reported from the Prophet or his companions.
If leaving this act causes misunderstanding then doing it is better.
Allah knows best.