yes of course , you can use the google+ API and the facebook sdk in your API. you begin by Login your users by googles+. in the end, if the user wants to share anything in facebook you authenticate him with facebook sdk, then ask for permissions then publish what you want. you can find many tutorials and samples about the authentication in google+ and facebook.
I don't want to give advertisements about libraries (and I didn't even test anyone of them...) , I hear about GigyaSDK. they have a good documentation of integration.