No, copies of a spell do not trigger Compost because they are not cards.
Copies of spells (and tokens) do go to the graveyard and can trigger all applicable abilities by doing that. They cease to exist the next time state-based actions are checked, but them changing zones is still a game event that happens.
However, Compost does not apply because Compost only triggers on black cards going to the graveyard. Copies of a spell are not cards, they're only objects without a card representing them.
706.10. To copy a spell, activated ability, or triggered ability means to put a copy of it onto the stack; [..] A copy of a spell is itself a spell, even though it has no spell card associated with it. [..]