The recently released SuiteCRM 7.3 has the ability to show desktop notifications and notifications on the site (in the header).
How can I add or trigger my own custom notifications through code?
~Elizabeth
It depends on what you where you want to do it. There is a JavaScript class Alerts that has a couple of prototype methods. If you open jssource/src_files/modules/alerts/alerts.js it has commented code on how to do this. You can also use the modules/alerts/alerts.php to create alerts that way. Just a heads up there are pretty big changes coming to alerts for 7.4 so you might want to hold on for the release.