What do you think Qui-Gon thought of Satine?
Hmmm... I think he thought of her as idealistic, given the fact Satine's father was a warlord and she was a Mandalorian pressing for peace. Mandalore hadn't seen peace in decades, and definitely in no one's lifetime (maybe Yoda's, but doubtful). Probably didn't agree with her political views for the most part, as he's seen the absolute best and worst of humanity over his five decades at that point.
But I think he probably enjoyed her banter and quick wit, and her ability to keep Obi-Wan from being a total hoe hothead.
I imagine that the times the trio were together, they probably discussed their respective cultures a ton. I can definitely see Satine being a history nerd - I mean canonically, she was studying on Coruscant when the civil war broke out on Mandalore (I don't have the exact source off the top of my head, someone help!) I personally think it's extremely important to have multiple perspectives of history, especially for a politician.
Jedi and Mandalorians have a long, intertwined history and I could see the trio overall really challenging each other's beliefs about their respective cultures. We already see that Satine is more than happy to share her voice, and that definitely came from an early age.
Sidenote - for Obi-Wan and Satine to develop the relationship they did, I believe that Qui-Gon was often gone for long periods of time during the Year on the Run, perhaps gathering allies or infiltrating various political factions, etc.