As much as I believe she would say that just to get on the MC's nerves, I don't think she would lie. She probably talked with Josy for real.
Then she arrives in the room and knows at once who she is.
Then that argument happens. Have you ever seen Quinn treat someone like that, so aggressive? Apart from Riona, nobody. As much as she didn't like what Josy said, Quinn wouldn't be shaken like that for so little.
And why would she refuse to allow Josy to join HOTs, saying that she wasn’t "HOTs material" when Josy clearly is? And just when she is in need of all the cats she can get. So she finally changes her mind when she finds out it's Tommy's sister, and that part of her plan we don't know yet, but it's clearly more important to her than using Josy to make a hell for MC or vice versa.
About the jealousy itself... There comes the part I want to believe.
She can see Josy as a real MC "girlfriend". And, because she is falling in love with him now, and not knowing how to deal with it...