Today, I found information about my father-in-law and chatted with others. It's vague. Should I tell my mother-in-law?

I can't. They are all your relatives, so it is better to have more than one thing. Pay more attention at ordinary times. If your father-in-law has excessive behavior, you can remind your father-in-law: I already knew about you, and I didn't tell my mother-in-law. I hope you will pay more attention in the future. A clever father-in-law will appreciate you. Remember what I said!