Peel, core, and thinly slice the apples. Toss the apple slices with the lemon juice and 2 tbsp water to prevent browning.
Heat 1 tbsp of the olive oil in a large skillet over medium heat. Add a few handfuls of arugula, sprinkle lightly with salt and pepper, and move around with tongs for a few seconds until the arugula is just wilted. Transfer to a bowl. Add a little more oil to the pan and continue to wilt the remaining arugula in the same way. Set aside.
Drain the apple slices and pat dry on paper towels. Divide the Brie into eight portions and spread one portion onto a tortilla. On one half of the tortilla, arrange a few slices of apple and some wilted arugula. Fold over the other half of the tortilla and press together. Repeat with the remaining tortillas, Brie, apples, and arugula.
Heat a little olive oil in the cleaned skillet. Put in a folded tortilla and cook over medium-high heat, pressing down with a spatula, until the base is brown and crisp. Turn over and brown the other side. Transfer the quesadilla to a cutting board. Cut into three or four wedges and keep warm. Repeat with the remaining quesadillas. Serve warm.