I had Genji give it to me as the result of an exchange he proposed via the run up calling your name. So he didn’t run up just to give it to me but it wasn’t part of the normal daily gift exchange. From what I’ve read you are still able to receive the photo from villagers in exchange for favors, if that is what you mean.
But, that thread also has other tips- I recommend the "stack" method they detail there, where the villager has to mail you a prize. I tried it last night and, since I've maxed most of my friendships now, I've already gotten THREE pics
It's never happened to me from them initiating a conversation, but I think WaileaNoRei is on the right track about an exchange needing to happen in order for them them give out their pic.