No. Well, you can say that but it would not be translation of the above. Your sentence in Czech would be "to jablko je moje". But just because you have an apple, it does not mean it is yours. Replace the apple with car keys. You happen to carry your friend's car keys, because she has no pockets or something. You have the keys but they, nor the car, is yours.
Maybe you're literally translating the Czech sentence as "the apple have I" and switching that around to make more sense in English as "the apple I have." That, as VladaFu points out, is just a phrase, while the original is a complete sentence. So, taking that into account, we read it as "I have the apple," because the alternative word order doesn't work in English. The placement of já at the end of the Czech sentence emphasizes that "I" have the apple, not someone else, as fehrerdef has explained.
UPDATE -- Strictly speaking, "The apple I have" could also serve as a complete sentence, but the word order is non-standard and, apart from use in the context of poetry or song lyrics, it sounds quite unnatural.
Well, in English you can leave out relative pronouns if they denote the object of the relative clause. So "The apple I have" means the same as "The apple that/which I have."
But of course this is not what the given Czech sentence says, but would indeed be "jablko, které mám já".
(and of course, in that case it is not a complete sentence, but nobody expects Duo to only treat complete sentences).
But maybe the OP was only puzzled by the word order. In English word order is rather strict. So apart from an interpretation including a relative clause "The apple I have" is not an option as an alternative for "I have the apple". In Czech, however, word order is much more flexible, so you can put the words you want to particularly stress in specific positions, usually at the end of the sentence. And so, instead of the neutral "(já) mám to jablko" you can say ""to jablko mám já", which stresses the "I", like saying "I have the apple" or even "It's me who has the apple" (and not another person).