Place the lemons in a large bowl of cold water. Cut a small amount of the very top and bottom off the artichokes. Snip the sharp points off of the leaves with scissors and rinse the artichokes. Cut each artichoke into quarters vertically. With a sharp paring knife, cut the thistle out of each quarter. Drop the artichokes into the water with the lemon to prevent browning.
Heat a grill to medium, about 350-400ºF.
Lay out a large piece of heavy-duty aluminum foil. Remove the artichokes from the water, shaking off the excess. Place in the center of the foil along with the lemon wedges. Cut the top off of the head of garlic and toss that in. Sprinkle with salt and pepper and wrap the foil tightly around the whole lot. Take another large piece of foil and wrap tightly around the packet so that some steam will build up while cooking.
Place on the grill, close the lid, and cook over indirect heat for 50-60 minutes.
Carefully open the foil packet and put the artichokes and lemon wedges on a serving platter.
Squeeze the garlic cloves into the mayonnaise and mix. Serve as a dipping sauce for the artichokes.