For one thing, technically the war against the dragons started with the Allagan's, whom are widely portrayed to be evil, for some reason. However, keep in mind it is stated that Midgardsormr and his brood came from the stars while the Allagans were already here on this planet. That's right. Midgardsormr and Bahamut, mostly Bahamut, basically invaded the planet and started carving out chunks of land for habitation. Yes, yes, what the allagans did with the dragons they captured is horrible, but I can hardly blame them for defending themselves against invaders.... least of all invaders from space.
Because Dragons have an immortal memory and the emotional maturity of a pre-schooler, they never let that grudge go, and distrusted the Elezen from the start.
It should also be noted that we do not know the whole story.
The fragments that the echo showed us came from Hraesvelgr's memories and a brief snippet that occurs after the battle with Nidhogg and then given a tiny summary of why they might have done it. It's assumed, post echo snippets, that the king Theodan coveted the Dragon's eyes and power, but there's no word on why, what gave him the idea, or how the actual battle occurred. The guilt of the surviving members of the 13 knight would seem to point guilt at Theodan's feet, but, even then, it's difficult to tell if they feel guilty because of what Theodan had them do, or because they couldn't save their fallen members from Nidhogg's wrath.
Also, keep in mind Nidhogg is clearly portrayed to have hated the Elezen from the onset and was merely waiting for them to give him an excuse to attack. The Hraesvelgr memory of eyeless Nidhogg makes it plain as day that Nidhogg was expecting the Elezen to be treacherous, telling Hraesvelgr that he warned him they would. He then goes so far as to complain that if Hreaesvelgr was not "enchanted" by Shiva he would have seen that the Elezen were enemies from the start. They made it sound as though he had been actively attempting to sway Hraesvelgr's opinion form the word go, and merely needed the right evidence. After all is said and done, Hraesvelgr simply takes Nidhogg's word for it, but the truth is that we don't really know what happened. The Echo did not show us the actual battle. All we have to go on is assumptions made after the fact. For all we know, Nidhogg could have orchestrated the whole thing just to create a catalyst for a war he wanted from the start.