Can you start birth control before your next period starts?

Yes, there is no method of Birth Control that requires you to wait for your period in order to start it. The current standard of care is to use the "Quick Start" method to get protection as soon as usual. Details about when to start and how long to use a back up method depend on which method you're asking about. You can confirm this information by checking references such as at ACOG or Contraceptive Technology.
Be sure to use a back up method, like condoms or abstinence from vaginal sex, for the first seven days of that first pack.