Thing is, I completed slimy business, but green skin quest doesn't appear, also slime king is not on character log, but I can already talk to him, I don't know if maybe I'm missing some conversation I have to do before green skin quest or maybe I have to wait x amount of days?
That’s pretty weird. From my test on a new save file, the quest still triggered normally with Slime King appear in the Log after winning him in first fight. The green skin quest won’t appear until you go to the Katarina Guild around 8 AM. When you meet the knight group for the first time there, the quest should trigger automatically. I also tested it with a new save, and it still worked.