The answer to this question doesn't really hinge on whether the travels are in Florida or out of Florida. But provided there is no existing order that adjudicates him to be the legal father, the short answer to this question is yes, mother can refuse to allow the father to travel anywhere with this child. Unless/until there is an order that gives father legal parenting rights, he technically has no rights to parent thischild.
That being said, if paternity is not really in question in this case (that is, father signed the birth certificate and/or mother will not challenge him being the biological father), then if I was the mother, I'd be cautious about making unilateral decisions regarding parenting, including father's contact with the child. The courts take it seriously when one parent makes sole decisions for a child. So while I think the law would side with mother on this issue, mother should be sure that she has a valid basis to refuse dad.
In the end, this is a pure parenting question: should a 2-month old be travelling away from mom during the period that dad wants to go? No one but the parents can really answer this question.
Before making any decisions, speak to an experienced family law attorney in your area to help you. Good luck.
I suggest you speak to an attorney in your area who has experience in family law. That attorney would be best able to address the specific circumstances in your case and to explain the options you have. The information above is for informational purposes only, and providing it does not establish an attorney-client relationship. Good luck.