No, this will not be an issue. As long as your name has been updated at the Social Security Administration then there will not be an issue filing the return.
IMPORTANT: You cannot file your taxes under your new married name until you have legally changed your name with the Social Security Administration.
You do not need to change your name on your W-2 in Turbo Tax. The W-2 will be matched to your return using your social security number rather than your name.