Your doctor has given you good advice. Nicotine in cigarettes closes down the blood vessels. This is called vasoconstriction. The result is a loss of oxygen and nutrition to the back. This includes muscles, tendons, ligaments, bones, and discs.
The risk of disc problems is increased in smokers. Smokers tend to have disc problems throughout the spine, not just at one or two levels. Some studies show that the number of cigarettes smoked each day does make a difference.
Quitting tobacco use is the best for your overall health, not just to reduce back pain. If you can't quit then cutting back as much as possible is the next best step.