No, Percy does not visit Montauk in "The Lightning Thief" book. The camp where Percy discovers he's a demigod is in Long Island, but it's located in the general vicinity of Montauk.
Montauk
to the underworld
In the lightning thief book, Percy does not know Luke is bad until the last chapter.
no Annabeth goes with Percy
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Camp half blood
He didn't, they added that into the movie but in the book I believe Percy never went to Nashville.
When Percy, Annabeth and Grover go to the underworld to meet the gods Luke come in the air with Hercules' flying shoes he has the lightning bolt in his and admits he stole it. Percy and Luke fight and the lightning bolt is safely returned to Zeus.
The name of the school that Percy attends in The Lightning Thief is Yancy Academy.
Percy is accused of stealing the lightning bolt so he has to go get it and return it to Zeus before he starts a war between the gods.
Percy Jackson's brother in the "Sea of monsters" book is, Tyson.
In "The Lightning Thief" by Rick Riordan, Poseidon and Zeus do not go to war directly with each other. They have disagreements and conflicts between them, especially regarding the protagonist Percy Jackson who is a son of Poseidon, but these disagreements do not escalate into a full-scale war between the two gods.