There is a strong odor of chalk in my home. We do not own any chalk or have seen any actual chalk in our home, yet the smell occurs randomly and is overwhelming. What could it be? Is it dangerous?

If you have drywall, you may have moisture or animals in the wal causing them to deteriorate. Drywall is gypsum encased in paper and glue -it is the most common form of interior wall. When it gets wet or is gnawed, the gypsum dust can be loosed. It smells like (and prettymuch IS) chalk.
Check your house of evidence of pests or moisture. You may also have mold.