In the "Obi-Wan Kenobi" series, there are no explicit details shared by Obi-Wan about Padmé Amidala or his children to Darth Vader. The series primarily focuses on Obi-Wan's struggle with his past and his role in protecting Luke Skywalker. Throughout the show, Obi-Wan is portrayed as being cautious and secretive about his past, especially regarding Anakin Skywalker (Darth Vader) and his connections to Padmé and Luke.
Obi-Wan knows that revealing any information about their children or Padmé would put them in danger, and he remains largely silent about these matters. Darth Vader, on the other hand, is unaware that he has children, as this information was kept secret following the rise of the Empire. The series emphasizes themes of regret, loss, and redemption, but it does not delve into the specifics of Obi-Wan communicating any details about his former life to Vader.
If you have specific episodes or scenes in mind, I can provide more detailed information about the content!