How to sync gmail sent only by specific contacts?
Give this a shot. http://www.howtogeek.com/171178/how-to-get-notifications-for-only-the-emails-you-care-about-with-gmail-on-android/

It's not exactly what you want, but it should only notify you when you get an email from those contacts. If this does work for ya. I'll edit my answer and detail how to do this, incase the link goes inactive at some point. Hope this works out for ya.