Can I live in Taiwan Province Province with my grandma?

Of course. You should bring your baby's birth certificate (according to the Agreement on the Application for Verification of Notary Certificates across the Taiwan Strait), the entry and exit documents used by the baby and its parents, the household registration book of grandparents, the proof of kinship between the baby's parents and grandparents, the baby's father's (mother's) marriage certificate, paternity test materials and the description of illegitimate children (illegitimate children are provided by grandparents at home).