Short answer: NO.
Long answer: copied shamelessly of Shog's answer in MSO:
This has been suggested many, many times, going all the way back to the earliest days of the site. Prior to the creation of Meta SO, feature requests were posted on UserVoice:
Comments were added to the system to allow users to communicate with each other in the context of a specific post.
Later on, the system was expanded to allow notifying other users of comment-replies.
Even with the ability to communicate with other involved users in regards to a specific post, the requests kept on coming. When Meta Stack Overflow replaced UserVoice, the number of feature-requests exploded:
The consistent response to this is that it's a bad idea, because:
It could hide information from the community: useful information transmitted privately is unavailable to other readers, subverting the core purpose of the site.
It could be used to harass other users ("Answer my question!", "Accept my answer!", "Yer momma so fat she overflows the stack!", etc...)
However, there is some value in being able to communicate with other users outside the context of a specific question or answer. This point was conceded with the implementation of a chat system: users on every Stack Exchange site can create and participate in chat rooms, integrated with the normal user accounts. There are still no truly private messages between users, but for those who want to chew the fat or discuss whatever outside the confines of the sites themselves, this can provide a viable solution.
Finally, it became apparent that moderators occasionally need to contact users about sensitive topics, and relying on email for this created problems. So the ability for moderators to send a private message to another user was implemented. The use of these messages are heavily restricted, reviewed by the system administrators, and reserved for critical messages only.
For normal users, the advice remains: if you want others to be able to contact you privately, add basic contact info to your profile's bio page.