Yes. I think you can do this. Below is the sample code. You have to use two events eventRender and eventAfterAllRender. or you can also hide the elements inside eventAfterAllRender.
eventRender: function (event, element, view) {
if( window.screen.width < 300 ) {
$('.fc-event-title').hide();
$('.fc-event-time').hide();
}
},
eventAfterAllRender: function( view ) {
$('.fc-event-inner').each(function(){
$(this).qtip(
{
content: $(this).children('.fc-event-time').html() + '' + $(this).children('.fc-event-title').html()
});
}
}
NOTE: This code is not tested. Change it according to your needs. Something like above will work for you.