Yes, you can paint your stained fiberglass door. All you need to do is wash it to remove any dirt or other contamination and prime the door. Sanding isn't necessary or recommended, but using a liquid deglosser is a good idea. For the primer, choose a good fast drying oil base primer like Zinsser Cover Stain.
After priming you can paint as usual with a good quality 100% acrylic paint.