You will probably find child seats quite few and far between in many countries, including Jamaica. Most airlines will allow you to carry a car seat for free as part of your checked baggage. Key word there is "most" and I'm not sure who you are flying with. Even if you do bring the seat, that's not to say the transfer coach will have seat belts!
You could find a private transfer company if you would like to ensure your childs safety but you will certainly pay a premium for that. A taxi at the other end is another solution.
You could contact Beaches directly and query as to what type of vehicle they use for their transfer service and if it is "car seat compatible" if you're worried.
This property has an excellent nanny service, I can't recall ever hearing a complaint about the child care staff or the kids club. The opposite in fact, it's usually glowing reviews.
Hope that helps!
Ginger ~ Blue Petal Weddings