What causes both of a person's eyes to dart from left to right a few times at once, then to go back to normal?
Assuming that these eye movements are involuntary a couple of things could be going on. Are you sure it is not an intentional movement? Extra ocular eye movements or EOM's are controlled by muscles that move the eyes. Cranial nerves, in turn, control these muscles. All muscles are controlled by at least one nerve.
The most benign cause of these movements could be a neuritis of the nerves that control the muscles of the eye. A more sinister cause could be that the person is having some sort of seizure like what is called an absence seizure. A third cause commonly seen in school age kids is a reaction to a stimulant they are taking. Many children nowadays take medications for ADHD, many (not all) of these medications are amphetamines. Sometimes these eye movements occur if the dosage of the medication is too high.
If these movements continue, I would suggest a good optometric exam first and then a followup exam with your family doctor. The family physician may refer you to a neurologist if indicated.