KeePassXC-Browser has an automatic update check, which will contact GitHub only so as to check for the most recent KeePassXC version. This check is optional and can be turned off in the settings. No additional data will be transmitted besides what is necessary to establish a connection.
The extension does not collect any information about user data or save it. When KeePassXC database is connected to the extension, and database is open, site URL is transmitted to KeePassXC for comparison. This is not saved or cached either. When saving new credentials, site URL, username and password are transmitted to KeePassXC. When disconnected or when database is locked, no information is sent to the main application.
The extension adds content to the page DOM when needed. However, pages cannot know the extension is used. It's only visible in the DOM when choosing custom login fields or showing the credential banner for saving new credentials. Icons, autocomplete menu and password generator are not shown to the page or its scripts.