This is the quickest method for preserving lemons that I've found.
fine sea salt
fresh lemon juice
Cut the lemons lengthwise into quarters without cutting all the way through. Sprinkle some salt into a clean jar and pack in the lemons. Put a lid on the jar, shake it up and freeze overnight. Sprinkle more salt onto the lemons and then press to release juices. Cover with fresh lemon juice and store in refrigerator for a week. Turn over or shake the jar every day.