CMMotionActivityManager does not currently offer a way to check authorisation status directly like other frameworks.
iOS - is Motion Activity Enabled in Settings > Privacy > Motion Activity
However, as the comments in the above question mention, if you attempt a query using
queryActivityStartingFromDate:toDate:toQueue:withHandler
and the user has not authorised your application, the handler (CMMotionActivityQueryHandler) will return this error.
CMErrorMotionActivityNotAuthorized