If your traffic is redirecting on your new domain from the indexed old domain, that's fine.
However, if you're worried about duplicate content because the old site has the same content as the new site, that can be a problem if the same page is being indexed on two different domains.
You can tell Google that your domain has changed in the old search console.
1. Make sure both domains are setup in Google Search Console.
2. Set up a permanent redirect on the old site's .htaccess file, redirecting every page to the new site.
3. Search "Google change of address" in Google. The first link above should be from Google Support. Click on that and they will give you a link to the old search console.
4. In the old search console, go to the gear on the far right and click on that. You will see "change of address".
5. Be patient. It can take months before those old URLs are gone from Google.