I'm not familiar with Zabbix, but there are two things you can do to check about the status of the components (Database, Web, Service):
If the database (SQL Server) wasn't working, or there was a problem with the website configuration, then visiting / will yield a 500 or some other not available message.
You can just query the ProGet service run state directly.
You could parse the text on the /connectors page or use the native API to get a status of the connectors.