It is imperative to confirm that the alertmanager configuration is appropriately configured to interact with SMTP servers before attempting to integrate Prometheus with a Outlook client for alert notifications. This entails supplying the alertmanager.yml file with the right smarthost, login credentials, and recipient information. Even with a suitable configuration, problems might occur, such as alarms firing and not receiving messages.

A thorough examination of the alerting setup, network settings, and confirmation that Prometheus and Alertmanager are both appropriately set up and updated are necessary for troubleshooting the problem of Prometheus alerts not triggering in the Alertmanager UI or being notified through Outlook. Two important configurations are 'prometheus.yml' for establishing scrape and evaluation intervals and 'alertmanager.yml' for routing and notifying alerts.