Is it permissible to bury an unbelieving child in Muslim graveyards if a Muslim adopted him and he died before reaching puberty?

It is not permissible to bury a disbeliever in Muslim cemeteries, whether he adopted a Muslim or not, or whether he reached puberty or not. A disbeliever is only buried in Muslim cemeteries if there is evidence of his conversion to Islam. Adoption is forbidden in Islam, based on the Almighty's saying: "Call them by [the names of] their fathers." (Al-Ahzab/5).