Is it possible to show only Google Fonts in the font drop down without using a 3rd party tool?
You can't do this natively in Photoshop no. This is exactly what a font manager is for.
I use FontExplorerX, which is great, but there are plenty of alternatives available and a number of previous Q&As about font management.
With a font manager you can group your fonts in to meaningful sets which you can enable and disable as you wish, so it should be as easy as a few clicks to enable your Google fonts and disable all others (although there will be some system fonts which you can't disable as your OS needs them to run).