The Facebook OAuth 2.0 implementation allows you to convert a client side short lived access token into a long lived token using the FB_Exchange_Token grant_type i.e.
https://graph.facebook.com/oauth/access_token?client_id={YOUR_CLIENT_ID}&client_secret={YOUR_CLIENT_SECRET}&grant_type=fb_exchange_token&fb_exchange_token={SHORT_LIVED_ACCESS_TOKEN}
My question is does Google’s version of OAuth 2.0 have a similar mechanism allowing me to get the short lived token via the client side flow, pass this token off to server and have the server convert that token so we can store the refresh token?
No, Google doesn’t have that functionality. You’ll need to use the server-side flow.
Can you give more info on the use case (in comments below?)