Ajax was born in Jade City to two FlameWing parents by the names of Adira and Rowan. His mom, Adira was a bartender at a tavern (I'm assuming these exist in Fallout Pyrrhia) in one of the busier parts of the city, and his father was a trader who made the journey between their home, Jade City, and Blackpoint regularly. His parents didn't see each other much, but they made the most of their time together.
Ajax was born into an average, middle-class Fallout family, and lived in an "apartment" near the tavern where Adira worked. Their apartment consisted of a main room, which had basic cooking things and a table, a tiny bathroom to the side, and a single bedroom that Ajax shared with his parents. It wasn't very nice or clean, as their building was more like a run-down motel. But, it was still home.
The young dragon grew up alone, dashing around the streets his mom proclaimed "safe" during the day with the other dragonets in the building. His mom worked long hours in the tavern, trying to get enough money to move to a home in a wealthier area and not have dragons living behind both walls, the floor and ceiling. Ajax almost never saw his dad, and was never able to really get close to him.
When Ajax was five, his dad left for a city even further than Blackpoint, one on the far point of the old SkyWing kingdom. He never came back. Ajax's mother was devastated when the members of his trading group came back and informed her that he had gone missing.
Ajax didn't know what to do. His father was gone, and his mother was slowly drifting away into alcohol and other distractions. Ajax found a job at as a guard at the entrance to Jade City, and he brought home a considerable sum to help him and his mom.
He cared for her the best he could, but nothing could stop the hallucinations that plagued her mind and caused her to lash out at her own son. Ajax couldn't bring himself to hit back, so he was scratched and bruised and hurt daily.