While Courts are unlikely to delegitimate a child, I do have some good news for you.
A small handful of foreign countries have worked out reciprocal arrangements where they will assist most US States in the collection of child support. Jamaica is one of those countries. Thus you should ask the Georgia Child Support office to help initiate a case on your behalf with Jamaica.
(Other countries that have similar arrangements are Australia, Austria, Bermuda, Canada (done by each separate province) Czech Republic, Fiji, Finland, France, Germany, Hungary, Ireland, Jamaica, Mexico, New Zealand, Norway, Poland, Slovak Republic, South Africa, Sweden, and United Kingdom.
The US State Department has previously commented that some state child support officials are unfamiliar with the procedures, and you may need to talk to supervisors. At least three other states, California, Texas and Illinois have handbooks on these type cases and you may wish to get one of those.
I was about to say "you can't terminate your son's father's rights". Courts are reluctant to end relationships between a parent and child unless the parent is either being replaced by someone else (adoption) or is such an extreme danger to the child that terminating the relationship is in the child's best interest. The fact that your son's father was deported is NOT a reason to terminate his parental rights. Additionally, your best method of protecting your children in the event of your passing would be to contact an Estate Planning Attorney for assistance in arranging provisions.
But based on your comment to Mr. Ashman's answer, my answer has changed...
You need to wait until you're married. Once you are married, your husband will have to file a step-parent adoption petition. Actually, he will need to hire an attorney to file the petition on his behalf. You attorney will request termination of the father's rights (or obtain the father's consent) as a part of the action.
Note, even if your fiance becomes your husband and then adopts your son, you still need to make plans for what happens if. Estate planning (wills, medical power-of-attorney, financial power-of-attorney, etc.) is something that everyone with family or property that they care about should take care of (and occasionally update)!
Good luck solidifying and protecting your family structure.
~ Kem Eyo
The above answer is a general explanation of legal rights and procedures. It does not constitute legal advice. Nor does it establish an attorney-client relationship between the individual posting the question and the attorney providing the answer.