I try to get alert data from some element using "metric_alert" API call, but I get no alerts in the respond.

I see alert is generated in the Metric Alert History though.

Only user who is subscribed to the alert rule in the Metric id 2268 (in our case) can get data about generated alerts with "metric_alert" API call.

Hence, you need to make sure that user who is used for "metric_alert" API calls (user who obtained token), is subscribed to the examined alert rule.

If he is not subscribed, you need to use another user for API calls who is subscibed to the alert rule.