I guess you could setup the alert emails to go to a box that triggers an SMS. Something along the lines of this:
Some telcos used to allow you to email to XXXXXXXXX(number)@operatorname.com to send this as a text message but maybe that stuff is not around anymore.
Maybe they have different API's and you could get the IT folks to set up some SMTP script to grab the incoming email and generate an appropriate SMS-request and pass to the provider.
Anywho, yes, possible, but not on the QlikView side.