Not gonna lie: Q popping in on Charles (Prince of Wales, Duke of Cornwall) hours into his reign as King of England the way Q pops in on Jean-Luc (Captain of the Enterprise) hours into his first mission is a fun story. I doubt Q would be as taken with Charles as he is with Jean-Luc, and given that the trial Q drags them to – the one in Picard’s past but our future – strongly resembles a scene out of A Knight’s Tale …
Let’s just say I don’t want the future of humanity to be in the hands of a member of a monarchy.
(Of course I know this is meant to be Cornwell, I just had to have a little fun because Q/the royal family is legitimately funny and terrible.)
I don’t ship Q with anyone. I think Q has a huge crush on humanity, most particularly Jean-Luc and Janeway, but he’s awful to everyone, even his favourites. He has zero boundaries, lives to embarrass and insult, randomly kidnaps people, and drags them into his own personal drama like it’s their job to make him a functional member of the universe but never once gives them credit for doing it. Remember when he turns Beverly Crusher into an Irish Setter for kicks? Don’t date that person.
And while Jean-Luc and Kathryn sometimes enable his behavior – because Picard loves a riddle and Janeway loves to flirt – Katrina would not. Katrina has an agenda, always, and she does not indulge arrogance. It would become abundantly clear almost immediately that Q’s goal is to get a reaction out of her and she would refuse.
She could, even, maybe, turn the tables and use her psychological skills to get a reaction out of him. He’d say she’s outmatched, and he’d be right, but she’d get closer than he’d like.
I don’t think Kat would become one of Q’s favourites. And I don’t think Q would intrigue Kat. Mainly, they’d annoy each other.