Ba has been linked with a move away to Galatasaray in recent days.
Quinn barked to the Evening Chronicle: "If he doesn't want to stay at Newcastle United and wants to be greedy, then it's got to be a case of bye-bye Ba for me.
"All the fans would like to know is one thing, and that's whether he's committed to staying at the club that signed him on a free transfer last summer and took a chance on him.
"His career is at a cross-roads in some ways.
"If he wants to achieve something and have a great career with Newcastle and become a legend along the way, he can.
"But if he wants to simply go and earn a bucket load of cash, then for me he's better off leaving now."