It is not in the Qur'an; it is a hadith. This can only be done if there is an Islamic State. There is none at the moment; hence, if so called Muslim countries do it then they must first establish an Islamic State, otherwise establishing half of the shari'ah is not Islam.
Moreover, Islamic scholars have unanimously agreed that the apostate should not be put to death immediately; rather he should be given sometime, asked to repent and must be called back to Islam kindly. This becomes even more important in the case of doubts and da'is (preacher to Islam) should help the person to resolve the doubts that the person has. However, if the person persists on apostasy then he should be put to death.
Imam ibn Qadaamah (May Allah have mercy on him) said in al-Mughni: The apostate should not be put to death until he has been asked to repent three times. This is the view of the majority of scholars, including ‘Umar, ‘Ali, ‘Ata’, al-Nakhaii, Maalik, al-Thawri, al-Awzaa’i, Ishaaq and others. Because apostasy comes about because of doubt, and cannot be dispelled in an instant. Time should be allowed for the person to rethink the matter, and the best length of time is three days. and Allah knows best
this is a gr8 response , although i am not sure about the 3 day window
i think some people have been given longer periods