gbif.org has no records for Melipona outside the Americas, and for Trigona it has 25 records with African locations out of 50,000 (the rest are all in the Americas).
From this evidence i think it is safe to assume that these two genera are not present in Africa or the Middle East, or otherwise, they have gone unrecorded by biologists for some reason.
The genus Trigona used to be applied more widely so there will be some records outside of the New World in the older literature. Trigona in the broad sense has now been broken up into several genera. Some people may not use the more finely divided classification so you may still see the occasional record outside of the Americas.