That would probably be possible if you include a link to your Contact Form 7 form plus all your input in the Form of the CF7 interface.
You’re right.
This is my form: https://lineadiretta.org/contattaci/
In CF7 interface, using default integration my input (in contact module) was:
<p>Il tuo nome (richiesto)<br />
[text* your-name] </p>
<p>La tua email (richiesto)<br />
[email* your-email] </p>
<p>Oggetto<br />
[text your-subject] </p>
<p>Il tuo messaggio<br />
[textarea your-message] </p>
[recaptcha]
<br><br><br>
<p>[submit "Invia"]</p>
Now, using plugin “Google No CAPTCHA reCAPTCHA Settings”, my input is:
<p>Il tuo nome (richiesto)<br />
[text* your-name] </p>
<p>La tua email (richiesto)<br />
[email* your-email] </p>
<p>Oggetto<br />
[text your-subject] </p>
<p>Il tuo messaggio<br />
[textarea your-message] </p>
[recaptcha recaptcha-222]
</br></br></br>
<p>[submit "Invia"]</p>
To see the module, I put in wordpress page:
[contact-form-7 id="2217" title="Modulo di contatto"]