If you do not have a juicer, tie the quartered quinces in a large piece of cheesecloth. Cook them in 2 cups of water over medium heat for 30-40 minutes. Add another 2 cups of water and 2 cups of sugar, and cook for another 30 minutes. Remove the cheesecloth bundle, squeeze out any remaining juice into the pot, and discard the fruit. Add the lime juice and simmer briefly.
Taste the syrup after cooking. Depending on the tartness of your quinces, you may need to add more sugar. Adjust to your preference before bottling.
Store the syrup in the refrigerator for up to 1 month. The syrup may crystallize slightly in the cold; simply warm it gently to re-dissolve the sugar before use.