How can I increase fertility naturally without invasive medical intervention?

Healthy. The best way is to stay healthy. To eat right and exercise, and to maintain a normal body weight. Also, you should do everything possible to avoid smoking since that has a large negative impact on becoming pregnant.
Monitor your cycle. By substracting 13-14 days from the length of a cycle, the day of ovulation can be estimated. In a 35 days cycle, for example, ovulation occured on day 21-22, a week late from the optimal day of 13-14. Stress can weaken the pituitary signals to the ovaries and cause delay in follicular growth and ovulation. Meditation or accupuncture can help reduce stress. Coitus near time of ovulation is best.