Sure, to importune is to beg, but use it only when you're talking about going beyond mere begging into more urgent territory. The woman importuned the judge to release her innocent brother from jail.
The original Latin term actually meant something closer to "to bother." Bear that in mind when you're deciding whether to use importune, because that's the kind of begging you'd want to be talking about when you do. Imagine needing something so badly that you cannot stop asking for it: then you might importune someone to get it. After a year of being importuned, dad let me have the car.