Yes, you can find hamburger meat in Mobay or Negril but it might not taste exactly like the meat you buy at home but for a good reason. Cattle are not pumped full of bulk-building steriods and are generally range-fed instead of a heavy diet of straw and hay.
You will find that Jamaicans generally do not eat much beef anyways. Of course you can always go to Burger King and satisfy your urges.
Peace and Guidance.