I haven’t used this one myself but AutoMattic has created a free add-on for Google Drive called WordPress.com for Google Docs, which can be added to any WordPress.com or Jetpack powered www.remarpro.com site. This allows you to send a draft over but does not allow you to publish to your site directly from the add-on. Much nicer than copying and pasting though!

The add-on can be found here.

The help section for the add-on can be found here.

Since the suggestion to link my wordpress with my google doc show up during when I was creating a blog post – I thought the suggested plugin must be developed by WordPress themselves.

So is there any plugin developed by WordPress on Chrome Webstore for this? Also should I be wary of installing 3rd party plugin?

I do understand you would be wary of giving access to your Google Drive or blog. The good news is that the one above was created by AutoMattic, the parent company of WordPress, WooCommerce etc. Hope that reassures you.

I have not seen the pop up you saw before but I seems quite likely that as this add on was created by AutoMattic, using this add on would have been the next step.