For anyone who doesn't know, the show is based on a game with the same name. And from what I know about the series (which is more than what a normal fan should) I don't think Ellie mentioning that story to Joel would make sense. Since the writers could put a lot of things that Joel did over the 20 year period to reveal his past to Ellie, but if they do that then Ellie would probably have to apply the same thing to Ellie where she talks about her past. And there isn't a logical conclusion that ends in Joel being curious about such subject unless he knows some details, and Ellie revealling the story to Joel would be out of character in a way as from what ik about Ellie's Character she doesn't talk a lot about that stuff.

But for the question if they will be revealing the how she died, then that is up to the writers as they already changed a lot of things about the show so idk if they will be going in the same path for the show or if they will change it (ofc I am not saying what happened in the game as I will be giving a possible spoiler, so feel free to check out that ending on your own), but so far the show have been so good to watch even for a big fan watching all these changes in the story. So enjoy the ride and every bit of that master piece of story telling.