Are all the volcanoes in the Caribbean connected so that an erupting volcano on one island will trigger the others nearby?

No, volcanoes in the Caribbean are not connected. Volcanoes on individual islands are formed by the same process, i.e. subduction at the plate boundary, but they do not share the same magma chamber, and are not linked by long underground magma conduits. A volcanic eruption on one island, therefore, cannot trigger an eruption on another island.