You can definitely mark an alert as Acknowledged. Goto the Alarms tab -> Select the alarms that you want to assign to a technician. Once an alarm is acknowledged the alerts will be suppressed.
For more help have a look at https://www.site24x7.com/help/alarms.html
Marking the monitor as green would falsely affect the SLA. Once you solve the problem and the monitor is up, it will automatically become available and your MTTR will also be the actual value instead of a false value. If you know the problem and are working on the solution the better way is to mark it as maintenance.
Another feature that you can use is to Automate the process by bringing in IT Automation.
Hope this makes sense.
Product Manager, Site24x7