Can a person get a head injury from moving their head back and forth very fast?

Other than look silly I don't see anything bad happening unless they bang into something.

Seriously doubt that you could move fast enough or with sufficient force to cause a head injury, assuming no specific contact with a solid object. But you might strain your neck, resulting in pain and stiffness.

Not at your age. In an old person there may be a risk of sub-Dural hematoma from rapid movement of the head. For healthy diet: http://www.choosemyplate.gov/., exercise 30 minutes/day, drink enough water daily so your urine is mostly colorless, have safe sex, no tobacco alcohol weed or street drugs. Get HPV vaccine. Wish you good health! Safe sex: https://www.hopkinsmedicine.org/health/wellness-an