I just figured this out, thanks to the code inspector in Google Chrome (if you’re stuck trying to figure out what CSS to modify, that code inspector is incredibly helpful).
Go to the General Options section for GLT, so from your WP dashboard go to Settings > Google Language Translator.
Within the Custom CSS field on the far right, enter this:
#glt-translate-trigger {
background: linear-gradient(#ea6044, #f97c63);
}
If you don’t want a gradient, just replace “linear-gradient(#xxxxxx, #xxxxxx)” with whatever hex colour you want.