And it is very much to his credit that I should think so.

Even though Darcy is so very reserved and "stupid" with the Hertfordshire people in general, he is quite surprisingly open and honest with his friends. He does talk about Elizabeth's beauty and his own admiration much more than I'd somehow expect. He doesn't keep it to himself, but he shares his feelings with the Bingleys with "great intrepidity" (ch.6). We cannot imagine him to be really best friends with Miss Bingley, still he tells her exactly what he thinks of Lizzy.
That's why I think that he must have told his beloved sister rather a lot of what's going on in his mind. He has pride enough - or rather self-confidence enough - not to be ashamed of his feelings. And I think Georgiana wouldn't have been so eager to be pleased with Lizzy if she hadn't known that her brother had very clear and decided intentions.