This page explains how to display and customize the reCAPTCHA v2 widget on a webpage. I have not used any existing node packages to implement it. There are so many Node packages which are much freindly to implement re-captcha on your website. My purpose not to use any node package is to get better understanding on verification of user response on server side.
I chose method which automatically render the reCAPTCHA widget. It is easiest method for rendering the re-CAPTCHA widget on webpage. All you need to do is to include the necessary JavaScript resource in head section and a g-recaptcha tag in form. The g-recaptcha tag can be used as DIV element with class name 'g-recaptcha' with your site key in the data-sitekey attribute.
This page explains how to verify a user's response to a reCAPTCHA challenge from your application's backend.
You can also defer render of widget by specifying your onload callback function and adding parameters to the JavaScript resource. You can find more info about it Google developer site. You can see the live implementation of it on contact page. Github repository for the project source code.