Praise be to Allah.
It is not permissible to bury a kaafir in the Muslim graveyard, regardless of whether he was adopted by a Muslim or not, or whether he had reached the age of adolescence or not. But if there was any indication that he was Muslim (or that he had become Muslim), then he may be buried in the Muslim graveyard. It should also be noted that adoption [in the sense that the child is given the name of the person who adopts him] is forbidden in Islam, because Allaah says (interpretation of the meaning):
“Call them (adopted sons) by (the names of) their fathers.” [al-Ahzaab 33:5]
And Allaah is the Source of strength. May Allaah bless our Prophet Muhammad and his family and companions, and grant them peace.