In Latin, 'fourth' (in the masculine form) is 'quartus'. It could have been used for a fourth son, I suppose, but we know surprisingly little about how the Romans actually spoke to their children so perhaps they did something else.
In British boys' boarding schools they used Latin for a series of brothers: the eldest Santos
boy would be Santos
until his younger brother joined; then you'd have Santos Major
Minor. Followed if necessary by Santos Tertius
Quade couldn't happen in Latin. But Quad- could, for instance the quadrivium was the group of four subjects that medieval kids had to learn at school, if they went. And we are familiar with quadrupeds, and quad bikes.
Of course, you did say you 'believe' that Quade means 'Fourth', and the point about belief is that if facts get in the way, you are free to ignore them - at your own risk, of course.
Your son has a beautiful name! I'm sure you could find something in there that you could call him.