Are there any shortcuts for a quick save?

As of Patch 1.1, Quicksave functionality has been added to starcraft 2.

Ctrl + Q will save the game.

Source: https://news.blizzard.com/en-us/starcraft2/842870/patch-1-1-0-now-live

As far as I can tell from searching and checking the hotkey options there is not key for this, however a quick key combination to save game would be:

F10-S-Enter

That brings up Menu, selects Save, and Enter submits the save file with the default chosen name and returns you to the game.

It's pretty fast see: http://www.ryancalderoni.com/archive/quick_save.swf