you can hide recaptcha badge in this way
https://contactform7.com/faq-about-recaptcha-v3/#hide-recaptcha-badge

to show the badge only in the contact form page it’s quite complicated, but you can try something like this:

.grecaptcha-badge { visibility: hidden; }
body.page-id-1234 .grecaptcha-badge { visibility: visible; }

replace the 1234 with the id of the page with the contact form