Please go to Front End PM PRO > Settings > Emails > New message content (click “Text” tab in Editor)
Then replace Message URL: <a href="{{message_url}}">{{message_url}}</a> with

<center><a href="{{message_url}}"
style="background-color:#49a9ce;border-radius:px;color:#ffffff;display:inline-block;font-family:sans-serif;font-size:13px;font-weight:bold;line-height:53px;text-align:center;text-decoration:none;width:200px;-webkit-text-size-adjust:none;cursor:pointer;">View Message</a></center>

You can do same for “Reply content”.
It will create a simple button, clicking that will go to your website message page.