The solution was to set the text browsers setTextInteractionFlags() to (textInteractionFlags() | Qt::LinksAccessibleByMouse).
In my case:
textBrowserReadConversation->setTextInteractionFlags(textBrowserReadConversation->textInteractionFlags() | Qt::LinksAccessibleByMouse | Qt::LinksAccessibleByKeyboard);
I found the solution here:
https://forum.qt.io/topic/70075/qtextbrowser-anchorclicked-not-emitted/2
But I'll leave the topic, because it was hard for me to find it through google or the search engine on this forum.