Since this is a custom monitor, you should be able to disable alerting by doing the following:
In the authoring tab of the console, locate Management Pack Objects > Monitors
Locate the custom monitor (you can use 'look for:' if necessary)
Right-click the monitor and select properties
In the properties window, click the 'Alerting' Tab. You can ckeck the box to disable tohe alert.
You may be asking to simply stop notifications without afftecting alerting. This is more difficult because your monitor is associated with groups and alert criteria. Your custom monitor is creating an alert that meets the criteria in a subscription. You need to change the alert to a type that doesn't meet the subscription's alert criteria setting. Otherwise, you need to change the subscription. However, this will impact all associated alerts. Take a look around the properties of the subscription for some other ideas.
I personally wouldn't mess around with creating new groups and the like for a single alert. However, that's an option as well.
