Whether you should exercise with sore quad muscles or not depends on the cause of the pain. For example, delayed onset muscle soreness (DOMS) is a natural response that occurs with a new exercise routine or increase in intensity of exercise, and as long as you have rested your quads for at least one full day, it is safe to exercise with DOMS. Doing so can even help reduce the DOMS as it encourages blood flow to the muscles which aids healing. However, if you have an overuse injury such as a strain, you should rest until this is healed.