Company I am with does not. I have worked for other companies that did, but generally it was considered a pittance. Min was like $500, max was like $3000. Most people offered the person they were referring half the money as an incentive.
Companies would push the issue, telling you this a chance for your friends to work with you.
I tended to consider the offers an insult, when you considered 1) the were saving 20 to 30K per candidate, as that was what the head hunters charged. 2) headhunters do not actually know the person so they are not really vetted the way a friend would be vetted. Because who would recommend a friend who would not be a good worker? A headhunter will all day and night.
But I suppose if they offered a good incentive, people would actually recruit ANYONE not just qualified people.