House Targaryen is an invading faction in A Clash of Kings. It is ruled by King Aegon Targaryen.
The Targaryen invasion of Westeros is no longer a quest base event and an invasion day is not confirmed. Most reports of their arrival fall between day 200-300.
The quest The Prince and the Queen has been removed in the current version (7.1).
Looking for Illyrios Manse near Pentos will give you nothing but frustration (or some decent loot from Pirate camps).