Could arthritis in my spine cause pain in my hips and knees?

Yes. When you have joint abnormalities in one place, you tend to compensate in other joints which then leads to abnormal wear and tear in other locations. Also, arthritis in one spot can indicate you are at risk in other places. Remain active, keep your weight down, and enjoy activity.
At your age unlikely. But you may have a systemic arthritic disorder that is affecting multiple joints so you need to see a physician for an evaluation.