The variables in the logmon probe allow you to read values in the log message lines to use in alarms or QoS. They do not help you count the number of times a specific log message was seen.
From what I have seen in the logmon probe, you can enable QoS for the number of times a log message was seen, but it does not look like you can use that to determine when alarms are sent.
Unless I am just missing the way to do this in the logmon probe, you may need to use some scripting in the NAS to achieve the desired result. You could make the alarms from logmon invisible and have a script to check if the suppression count on the alarm is at least 59. If it is, the script can generate a new alarm that would be visible. Catching when the issue has cleared is a bit trickier, but I am sure there is a way to do it.
You could also use the logmon probe to post custom messages to the bus rather than send alarm messages. You would probably need to write a custom probe to process the messages, so I doubt this is any simpler than using a NAS script. It may actually be simpler to write a custom script to scan the log file rather than use the logmon probe. (Of course, the logmon probe is good at remembering where it was in the log file on the previous check, so there may still be an advantage to using it.)