In The Hobbit, on what special day do Bilbo and the dwarves discover a way into the Lonely Mountain and how do you know?

We are given the answer to this question at the end of
Chapter 3, when Elrond helps them work out where they need to go and tells them about
moon runes. Reference is made to "Durin's Day," which pleases the dwarves incredibly
because Durin is actually an ancestor of Thorin and a very important person for dwarves
in general. Note what is said about Durin's day:

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"The first day of the dwarves' New Year," said
Thorin, "is as all should know the first day of the last moon of Autumn on the threshold
of Winter. We still call it Durin's Day when the last moon of Autumn and the sun are
in the sky together."

So,
reference is made to the importance of Durin's day, however it is not clear in the novel
if they actually discover the way in to the mountain on Durin's Day itself - no mention
in Chapter 4 is made of any particular day except for the fact that winter is coming and
it is getting much colder.