Hello my name is Alex an I will try to help you with this matter.
A copy of your birth certificate can be obtained from the Registrar General's Department. The easiest way is probably by applying on-line at www.rgd.jm. There is unfortunately a fee to pay, I believe that this is in the region of J$650, for the fist copy and J$100 for the second.
I hope this answers your question, if I can be of further assistance please contact me. If you are happy with this answer, please remember to rate it. Alex.