I m a Nigerian and I want to spend my two weeks holiday in Jamaica with a tourist
You can fly to Montego Bay from Manchester, London, Brussels, Duesseldorf, Frankfurt, Munich, Milano and Verona and few more non-US locations in the Caribbean and central America. Also a dozen airport in Canada serve Montego Bay.
Most of the connections from Europe are charter/budget carriers, so you may have to split this into two trips.
A casual search showed non-US connections through Toronto (YYZ) with Air Canada, Egypt Air, Turkish, etc on a single ticket.