How did the war between Predators and the Aliens start?
As per Alien vs. Predator (2004), every 100 years the predators come to Earth to hunt aliens (xenomorphs). The predators use humans as a way to harvest xenomorphs. This has been happening for thousands of years.
Does anyone know how the war between predators and xenomorphs started? Is it mentioned in any predator movies or comics?
It's not a war. Predators hunt xeno because they like hunting. In the first issue of Aliens vs. Predator we can see that predators keep queen and then send her egg to different planets (chosen by predator) to create hunting ground.
Because of the constant change in settings between movies and comics we can only assume that Predators been hunting for something for a very long time and when Aliens came to existence they switched to hunt mainly for them. Using humans as a host because humans (until movie Alien3) turned into Warriors. In Alien3 they established that xenomorphic take some characteristic of their hosts.