Yes, I would certainly eliminate the articles if I were using the name "Poseidon's Trident" to identify an organization. To identify the object (the trident) itself, I'd want to leave them in, however. Awkward little beasts though they sometimes are.
Come to think of it, I'd also eliminate the articles if it were the title of a work of art, book, poem, etc. Primalpharaoh calls his picture a "portrait." Hmmm.