Hey there @Sauraus , after researching the challenge internally with our support team it appears that at this moment, it is not possible to implement Google Recaptcha in Hosted login. However, you can implement the Google Recaptcha by redirecting users from rule during authentication flow.
https://auth0.com/docs/rules/current/redirect
Sample Code:
redirect-rule.js
function (user, context, callback) {
// this rule requires the following configuration values:
// CAPTCHA_SECRET: a 32 bytes string that will be the shared secret between
// the rule and the webtask
// AUTH0_DOMAIN: your auth0 domain (e.g. account.auth0.com)
// CAPTCHA_REDIRECT: the URL for the webtask that will show and process CAPTCHA
// Put a specific client ID if you dont want CAPTCHA for every client
// if (context.clientID !== '[your client id]')show original
webtask.js
'use latest';
// This webtask requires the following configured secrets:
// RECAPTCHA_SITE_KEY These two values can be obtained from http://www.google.com/recaptcha/admin
// RECAPTCHA_SITE_SECRET
//
// CAPTCHA_SECRET A 32 byte string that is the shared key between the rule and the webtask
// AUTH0_DOMAIN Your Auth0 domain (e.g. account.auth0.com)
show original
Please let me know if this helps you on your quest! Thanks!