This is the 4052nd most frequent Polish word.
zielonego
"Zielonego" means "green" (genitive or singular masculine/neuter form) in English.
In this context, 'zielonego' is used as an adjective in the masculine singular accusative case, describing the color of the noun 'cat' (kot).
Mam zielonego kota.
I have a green cat.
Here, 'zielonego' is used idiomatically as part of the phrase 'zielonego pojęcia' to express a complete lack of knowledge or understanding about something.
Nie mam zielonego pojęcia.
I have no idea (I don't have the faintest idea).
In this case, 'zielonego' functions as an adjective in the neuter singular accusative case, modifying the word 'jabłka' (apple) to denote its green color.