A fair bit of this is hearsay, and not possible for facts or stats on the proposed outcomes. Just chewin' the fat.
If you're going to do this it would be likely worth doing for questions (accepted awarded) and answers (accepted received).
So questioners are actually awarding and answerers working towards them.
If you only create a badge for answerers, then questioners may be even more reluctant to accept an answer if they know the answerer could get a badge.
Whether you do one or the other or both, it's likely going to bring a fair amount of unsavoury deeds.
I'll address both..
Users wanting the badge might accept answers they might not have otherwise done, even if they're poor quality answers.
This sends the message to other visitors that "this answered my issue", and causing confusion when you look at the question and answer and wonder how it answered it.
There's a danger badge hunters will ask questions just for the sake of being able to accept an answer.
Poor answers then obtain some bogus rep as the questioner accepts any answer given.
The potential of generating bad questions, and poor answers being accepted, is not a good thing at all.
Users may attempt to answer more questions attempting for the badge. If they answer trying for a badge, it's possible they write a good one, but I think more badge whores will just write any crap, introducing more rubbish answers.
While we have the means to deal with this, it's time wasted for people voting, flagging and commenting. Especially if it's not a real question requiring a resolve or advice for someone and is just badge farming.
Users wanting the badge will mark more answers as accepted, and perhaps continue the habit once earned, which is a good thing if a good answer.
There's usually at least a half decent answer to mark as accepted, and usually by 10 mins good answers have upvotes and bad ones downvotes, so even badge gamers who wouldn't have otherwise accepted an answer will likely choose the better one.
Users may try a bit harder to give good answers to gain the badge.
Personally speaking, I think we have plenty of badges!
Certainly enough that most areas of expertise, interest and skill are widely covered so most people can gain a fair number of them without too much trouble.
Currently I believe (mostly) questioners only accept answers if they feel it was worthwhile, and sometimes even then they don't bother.
While this isn't ideal for people answering wanting their accepted rep and green tick, it works well as only decent answers get accepted (mostly).
Plus answers not being accepted is no where near as bad as poor answers being accepted.
People wanting to earn yet another badge aside, I think this would have a detrimental effect, and I'd rather see dev time spent on other things anyway.