Generally, no. Unlike president which is capitalized when referring to a specific president, such as President Obama, first lady shouldn’t be capitalized. According to the White House Museum website, it is acceptable to honorifically capitalize first lady such as “First Lady Michelle Obama.” However, since the first lady is never officially elected or sworn in, “first lady” is not a title and shouldn’t be capitalized.
We guess that FLOTUS should instead be flOTUS. 🤷♂️