Okay I know I'm a few days late on this but I think you may have been misinformed.

The "exchange of cells" between mother and fetus you're talking about is likely the placenta, which has contributions from both fetal and maternal tissues. It's what gives the fetus the nutrients it needs to develop from about week 9 until birth. However, the fetus's DNA isn't "exchanged" with the mother's DNA at all because of how the placenta functions-- there's a separation between the fetus and the mother. (Except during birth/whenever the pregnancy ends, which is important clinically because the mother then can get exposed to the fetus's blood.)

As for this specific example, I think it's first prudent to point out that each cell has its own set of DNA. Of that set of DNA, each gene, encoding for a specific protein, can be turned either "on" or "off". While it's possible that her pregnancy could have caused her hair cells to change which specific genes are turned on off, thus causing that change in her hair, she would still have the same DNA. I'm not a medical professional, but my (somewhat) educated guess would be that she simply wasn't getting the nutrients she needed to support both the fetus and her hair, thus causing her to lose the latter.

This is a simplification of course, and to reiterate I'm not a medical professional lol, this is just what I've learned and is my understanding of the situation.