It sounds like you are getting a lot of monitor wash either directly into the room and or bleeding into any open mics.
As just mentioned check your overall stage monitor sound and maybe work with the overall stage volume to run the monitors at a lower level, in the long run that would help everything!
Now to the original question, yes you can set up all of the monitor mixes to be on a mute group, I actually take it one step further a lot of the times, I’ll put all of the monitor mix outputs on a DCA group so I can I can turn them down some in between songs for speaking, you can also do a hard mute via the DCA.
If you have people coming up to speak who are all ready half afraid to talk on a mic and then they hear themselves in a stage monitor more often than not they will back away from the mic, turning the monitors down or off can really help in that case as well.