Peel the shrimp and cut them each into 3-4 smaller pieces.
Thinly slice the green onions and core, seed and mince the jalapeño peppers.
In a medium size bowl toss together the shrimp, green onions, jalapeño pepper, cilantro, lime juice and salt.
Make the quesadillas by scooping some of the shrimp mixture onto half of each soft tortilla.
Evenly sprinkle some cheese onto the shrimp mixture and then fold the soft tortilla over covering the shrimp and cheese. Do this with all the quesadillas.
Heat grill to medium high heat, around 400-450F.
Carefully place the quesadillas on the grill and close the lid. Cook for about 6-8 minutes, or until the tortilla is starting to brown and the cheese is beginning to bubble.
Note - A grill stone or pizza stone works great for quesadillas. If using a stone, make sure it has time to heat up before placing the quesadillas on the grill stone.