Sounds like you are talking about the share/action icon; where it shows you AirDrop, a row of color icons, and another row of monochrome icons below that.
In order to add new share extensions (the colored ones) or action extensions (the monochrome ones) you scroll all the way to the right then tap the three dots icon. You can select and rearrange extensions there.
Extensions are installed by apps, meaning they don't just show up out of nowhere. For instance you must have Evernote installed before you will see the Evernote extension in the share actions.
To the best of my knowledge there are no Google actions, at least not yet. I've got most of the common Google apps installed, including Google Search, Google Drive, and Google+ (which I would expect to be the first).