A few might correspond to isotopes found in the Island of Stability, but there simply isn't enough room there to account for the hundreds of extra elements present in the Star Trek universe.
There is enough room, because these new elements invented for Star Trek aren't in addition to the existing elements from the real world - they replace many of them!
The Star Trek periodic table described by Memory Alpha based on images seen in the TNG episode Rascals is quite weird and unfamiliar to real chemists:
Elements are sorted by atomic weight, which is seemingly unrelated to atomic size and doesn't correlate (in the cases of elements that do exist in the real world) with real atomic weight.
In this table, though, nearly all the entries are joke elements, so perhaps we can safely ignore this as "just an in-joke" (don't kill me, all you hard-core Trekkies!) The Starfleet Medical Reference Manual has a more sensible-looking periodic table with far fewer imaginary elements:
In either case, given that there is still a periodic table, and its overall shape is similar to that of the real one (although some of the elements have different names), it's reasonable to conclude that atoms of different elements are made up of varying numbers of protons, neutrons, and electrons just as in the real world.