The Quran doesn’t allow killing apostates (riddah, abandonment of Islam).
According to the Old Testament the apostates have to be killed (Deuteronomy 13:6-10). The traditional Muslim understanding asserts the same punishment.
However, according to the Quran the apostates will get their return only in the hereafter (al-Baqara 2:217; Al Imran 3:85-90). The Quran doesn’t allow a punishment for them in this world. Anyway, there’s no compulsion in religion (al-Baqara 2:256; Qaf 50:45). The humans can not punish a crime, which has not been committed against them. Faithlessness and apostatize concerns only the Creator, not humans.
“Those who believe, then disbelieve, then believe, then disbelieve, then increase in disbelief, God will not forgive them, nor will He guide them to a way.” (an-Nisa 4:137). This verse tells us about belief after apostatize. If the apostates have to be killed, how could they believe again? Thereby the Quran definitely doesn’t allow killing apostates.