Remember that when Darcy tells Lizzy his sister wants to meet her, Darcy wants to introduce the two; so Georgiana only had to express the desire once in passing for Darcy to use that excuse. What might Georgiana have heard to prompt her to say, "That's someone I'd like to meet"? Choose any one of the following which I expect Darcy might have written (pre-proposal) from Rosings:
A young lady I had met in Hertfordshire is here; her love of music reminds me of you.
Cousin Fitwilliam has been back to the parsonage often; I suspect he enjoys conversing with one of Mrs. Collins's guests.
The young lady seems totally at ease with our Aunt Catherine.
The young lady's conversation never ceases to amaze me, yet she speaks with such cleverness and humor that one seems not to mind anything she says.
You would enjoy conversing with Miss Bennet.
We know Darcy writes his sister, presumably with some frequency, and I expect the run-up to the proposal is the time he is most likely to mention something about Lizzy. I also think that Darcy's speaking "in such terms of Elizabeth as to leave Georgiana without the power of finding her otherwise than lovely and amiable" in Ch. 45 could have occurred after Georgiana's arrival at Pemberley, perhaps even on their drive back from the introduction. After Georgiana met Lizzy, Darcy could point out to his sister all the wonderful qualities Lizzy had displayed at the interview, and Georgiana may have asked her brother questions then as well.
I do not think Darcy confided in anyone that he had proposed to Lizzy, not just to save himself the embarrassment but to protect the feelings and reputation of the lady.