No, there is no feature on the site that allows you to do this.
Put yourself in the other user's shoes. Do you want anyone and everyone to be able to spam you with links to questions? I didn't think so.
The Stack Exchange sites are not social networks nor forums. There is no support for sending private messages to other users. That's completely by design and unlikely to change. (And if it ever does, you'll see a whole bunch of us leave. I'll be first. Form a line—single-file, please. No shoving.)
Of course, if a user chooses to disclose his/her contact information in their profile, you can use this to contact them. Or if you both hang out in the same chat rooms, you could post a link to the question.