When Percy is assigned his quest by Chiron, he is told that Zeus thinks that Poseidon stole his thunder bolt. Chiron tells him,
"Your father and Zeus are having their worst quarrel in centuries. They are fighting over something valuable that was stolen. To be precise; a lightning bolt." (pg 135)
Percy has just found out that his father is Poseidon. Poseidon has also broken a deal he made with Hades and Zeus that they would father no more mortal children after World War II. That is why he was reluctant to claim Percy as his son. Percy asks if his father would steal the lightning bolt. Chiron says,
"Most thinking observers would agree that thievery is not Poseidon's style. But the Sea God is too proud to try convincing Zeus of that. Zeus has demanded that Poseidon return the bolt by the summer solstice. That's June twenty-first, ten days from now. Poseidon wants an apology for being called a thief by the same date. ....Unless someone intervenes, unless the master bolt is found and returned to Zeus before the solstice, there will be war." (pg 137-138)
When Percy goes to see Hades in the Underworld, he finds out that Hades had his helm, the most powerful weapon of fear, stolen. He thinks Percy has it. Percy suddenly finds himself in possession of the lightning bolt, and he has no idea how it got in his backpack. He is having a hard time convincing Hades that he doesn't have the helm. Percy suddenly realizes,
"...I'd been played with. Zeus, Poseidon, and Hades had been set at each other's throats by someone else. The master bolt had been in the backpack, and I'd gotten the backpack from..." (pg 314-315)
Hades doesn't want war. He has his hands full with the people in the Underworld now. He says,
"Do you think I want more subjects?... Have you any idea how much my kingdom has swollen in this past century alone, how many subdivisions I've had to open?" (pg 311)
He does, however, threaten to stop death and release all of the dead souls back to earth if his helm isn't returned
The war is between Zeus and Poseidon, although all the gods would be drawn into it. Chiron tells of the consequence of such a war.
"Imagine the world in chaos. Nature at war with itself. Olympians forced to choose sides between Zeus and Poseidon. Destruction. Carnage. Millions dead. Western civilization turned into a battleground so big it will make the trojan War look like a water-balloon fight." (pg 138)
As previously mentioned, there is no specific quote. Also it can be inferred that Hades does not want such a war as he is tired of all the conflicts, and that his kingdom seems to become dumping grounds for the whole masses of the dead.
There are no specific quotes. More or less, Hades offers Percy his mother in exchange for the lightning bolt.
There is no specific quote available when he is starting the war. But, there are specific quotes available for percy in the story.