Lots of secondary sources say it is true - though it happened in 1794 when the Regiment was being formed. "she tempted each volunteer with the offer of a golden guinea (the king's shilling) from her lips."
from an article in the Sunday Times, July 13, 2003
It also appears in info from the Gordon Highlanders Museum
https://www.gordonhighlanders.com/the-gordon-highlanders/Quite a few mentions from the 1850s, but I can't see anything in newspapers from the 1790s. The earliest I can see is in 1853 (again, some time after the actual events):
"... Gordon Highlanders - so called for having been originally raised by the Duchess of Gordon - of beautious and witty reputation, when "George the Third was King", in consequence of a wager of £10,000 with the Prince of Wales, as to which of the two could raise a regiment first. The duchess levied and equipped 1000 men in three months from estates in Scotland, but it is needless to add - his Royal Highness not being a despotic monarch - could not raise a single man ..."
Friday, July 1, 1853
Publication: John O'Groat Journal
Another item from 1854 says a correspondent witnessed the recruiting, however again, I can't find the original reference.
Thursday, July 20, 1854
Publication: Inverness Courier