In a mixing bowl, combine the turkey and zucchini. Season with jerk seasoning and a generous pinch of salt. Mix gently to combine to ingredients without overworking the meat. Form into 4 patties.
Grill the burgers over medium-high heat, turning at least once, for 10-12 minutes or until cooked through. Top with cheese. Cover and cook 30 seconds to 1 minute to melt.
While the burgers cook, prepare the salsa. In a second bowl, combine the pineapple, scallions, cilantro, jalapeño, and lime juice. Mix well and season to taste with salt.
Top the burgers with the salsa and serve on toasted rolls.