Adj. is only a short way of writing the word "adjective" on the page. You see it in dictionaries, grammar notes, and vocabulary lists, almost never in a normal sentence. If a line says "happy adj.", it is telling you the job of the word in grammar, not giving you a new English word to say. Teachers and books use it to save space. Though actually, you would not read it out as "adj." in class as if it were a word of its own — you would say "adjective". It is a writing mark, like n. for noun or v. for verb.

「形容詞」的書面縮寫（字典、文法筆記裡用來標詞性）