Can we use API to send Desk Alerts notifications to users that automatically will close?

You can use the following options:

autoclose=integer; manualclose;

Set auto-close time in minutes. The alert will appear on the screen for the given time period and then it will disappear automatically. And "manualclose" is ability to close the alert manually in case of auto-close setting active.

You can read up on our API here:

http://alert-software.com/wp-content/uploads/2015/05/DeskAlerts_API_Manual.pdf