Even though the Lannister-Stark hostility kinda began when they were on the 2 different sides of the war and when Tywin betrayed the Mad King to gain entry and cause most houses saw the Lannisters as greedy and prideful opportunists, as opposed to the Starks' well-earned reputation of being honorable oathkeepers, I have my own au for the stark/lannister hostility and I really wanted to share it
So imagine if the hostility started when Aegon I Targaryen was king and (King) Loren I Lannister (the Last) was married to Lyanna Stark (my OC and (GOT) lyanna Stark was named after her) they married in a political marriage but soon fell in love. As years passed, they had a couple of sons, and as their sons grew up, Lyanna experienced some miscarriages which caused her to fall in deep depression. So when she got pregnant again and didn't miscarry, she and Loren were ecstatic till 9 months later. Lyanna gave birth to stillborn twins, which caused her depression to worsen. So a couple days after she gave birth, she jumped out a window falling to her death into a flower bed of winter roses. She was found by a servent, and before a letter could have been sent to the starks telling them about Lyanna and what happened, whispers started up. The whispers traveled all around the seven kingdoms, all saying she was pushed by Loren, while Loren who was overcome by his grief of his wife being gone, slit his own throat, and with that, the whispers grew, saying Loren took his own life so the starks wouldn't come after him for killing Lyanna. Sadly, the whispers made it to the Starks before a raven could, the starks started believe the whispers the more they heard them and even though they started to hate the lannisters, they still granted Loren's wish of being buried in the winterfell crypts next to Lyanna, knowing that's what Lyanna would have wanted. In the casterly rock castle, they hid away all of Lyanna and Loren's things, putting them in their bedchamber, the doors closing never to be opened again and their story to never be talked about again. With Loren's and Lyanna being gone, their sons slowly become heartless, greedy, and prideful opportunists, causing the rest of the lannister family to be the same. Over the decades, the hostility between the Starks (who still believe the whispers) and the lannisters only grew.