The difficulty in this situation is that during the early months of the child's life, the visitation to the non-custodial parent is typically limited. During these months are when the child bonds with the mother. You can file for paternity determination and custody order. However, any type of visitation orders will be very limited, although they may be gradually increased.
There will also be jurisdictional issues depending upon where the child is born.
A 730 move-away evaluation may also be in order depending upon the timing of said move.
You will likely need to consult an attorney further regarding this issue, sooner rather than later.
Disclaimer: This answer does not constitute legal advice, and should not be relied upon as such. Each situation is fact specific and therefore any legal evaluation may vary. Please note it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the relevant facts and documents at issue. This answer does not create an attorney-client relationship.
Sign up to receive a 5-part series of useful information and advice about child custody law.