That should be possible. A status field in the record can change from one value to another when someone marks it "Read". This can literally be the text "read" and "Unread" or you can use other values.
But then you can set up a relationship that matches only to those records where the value is set to "unread". (and an OnRecordLoad trigger on the layout where the user views the records can change the field to "read" automatically.)
Phil, I believe the OP was looking for a way to show a badge on the FM GO icon. An alert much like an email alert on a IOS device. As far as I know there is no way to accomplish this.
Hmm, hadn't thought of that and no there is no way currently to do that....