The daughter is five years old. OP says they are introverted, I don't know if that is just her and her husband or the little girl as well. Sounds like the mother-in-law is more extroverted, maybe that's why they don't get on. If the little girl is extroverted, like her gran, she probably loves spending time with her, she may think she's fun and exciting, outgoing and sociable, all things that introverted people are not. There's nothing wrong with being introverted, but assuming this means quiet, shy, unsociable, uncommunicative, it is probably a pretty calm and peaceful household. If the 5 yr old is happy with this it's fine, but if my almost 3 and 5 yr old granddaughters are anything to go by, they love a day with me, having fun, going crazy, boisterous and laughing their heads off 😂🤣.
I'm an only child and was brought up in adult company, I couldn't relate to other children who were boisterous, noisy and mischievous. I now enjoy nothing more than acting the goat with my granddaughters, seeing how high I can get on the swing, letting them stick stickers all over my face, singing songs at the top of our voices and dancing on the patio 💃🎶🎵.
I really hope OP's daughter gets to enjoy the company of her gran, it's a completely different relationship to that of parents. Most of us no longer have the worries and concerns of young parents, we've been there, done that, and come out the other side. I don't feel as strung up as I did thirty years ago, I am comfortable in my own skin, I feel more confident and relaxed.
Don't worry, be happy 😁.