Izuku was quirkless before, but then he got One For All. His mom saw him use his quirk in the Sports Festival, but she never asked him how he got his quirk. And Izuku told All Might that he hadn't told anyone about One For All, not even his mom. Yet, his mom never asked him about his quirk. Why didn't she say anything?
I can think of a few possible reasons why this would have happened.
Izuku's mother assumed that he developed his ability at some point, but chose not to use it often due to its side effects. In this case, she may not have mentioned it, realizing it may be a sensitive subject.
Izuku has a very close relationship with his mother, and people with these types of relationships can often tell a lot more about each other than most. His mother likely sense that, for whatever reason, he didn't want to talk about his quirk, and decided not to press the issue.
She asked Izuku offscreen, and he deflected her question somehow; which would end up resulting in the same effect as 1. or 2.
This is about Dekus mom knowing about the quirk I just watched season 3 episode "From Iida to Midoriya" and whilst on the train Deku has a flashback. Earlier at the hospital he was on the phone with his mom and she asked him do you have to go back to UA. We don't hear what he says until this flashback. He says mom All Might gave me everything, he trusted that I'd make the most of his quirk. This happens at just over 15 mins into the episode. I've watched this series many times and it's the first time I caught that confession. If you go back to the part where he's on the phone with his mom then fast forward to the confession he's still got the cellphone to his left ear and wearing the same clothes. The part that is confusing is a later episode when they talk to his mom , all might does say he's his chosen successor to be the next symbol of peace but she seems clueless to how he got his quirk. So I'm confused, The only other possibility is it's just his thoughts and it wasn't a real conversation, what do you think?