Can I use a Google badge for my Google+ profile pic?
Bear in mind that the google guidelines for actual, main profile photos are very strict. Specifically, they must be a clear HEAD SHOT and not offensive. If you dont follow those guidelines, it will be removed, and more importantly will never be used to indicate authorship in the google results.
In short: no, you cant.
Yes you can.
Make your profile pic with google plus badge on -
Below is the site mentioned in the above video -