blue in the face - Phrases/Idioms
The American Heritage® Dictionary of Idioms by Christine Ammer. Copyright © 2003, 1997 by The Christine Ammer 1992 Trust. Published by Houghton Mifflin Harcourt Publishing Company. All rights reserved.
Exhausted from anger, strain, or other great effort. For example, You can argue until you're blue in the face, but I refuse to go. This expression alludes to the bluish skin color resulting from lack of oxygen, which presumably might result from talking until one was breathless. See also under talk one's arm off.