I have never heard this usage myself, but as @choster says in the comments, it has been/is used by some to refer to the Old Testament of the Bible, as opposed to the New Testament.
It seems to be an infrequently used idiom with that meaning in print and general broadcasting (for example, in news and politics broadcasting), to the extent that may be gauged by searches of general English-language corpora. Perhaps higher usage would be found in specialized searches of Christian religious broadcasting and writings.
(I am new to searching using the online gateways at corpus.byu.edu. Perhaps someone else with more experience can confirm my findings?)
searching for "THE OLD BOOK" finds:
British National Corpus - 3 total hits; 2 of those completely irrelevant
1 identified use is in:
all over the country clergymen continued to conduct services based substantially on the old Book of Common Prayer.
(Note that "the old Book of Common Prayer" kept turning up in all of the searches I ran of the various corpora.)
Corpus of Historical American English - 46 hits, some duplicates
only 6 uses seem to be relevant:
Bad as the old book of Genesis is said to be, it is certainly better than the new.
^ 1867, in Atlantic Monthly
It shows what is needed for a city man or woman to support a family on the proceeds of a little bit of land; it shows how in truth, as the old Book prophesied, the earth brings forth abundantly after its kind to satisfy the desire of every living thing.
^ 1907, Three Acres and Liberty, by Bolton Hall
Ruth's story is one of the most beautiful ones to be found in the Old Book.
^ 1915, Good Cheer Stories Every Child Should Know by Chelsea Curtis Fraser
" All flesh is grass, " says the old Book.
^ 1922, The Last Harvest by John Burroughs
The old Book says that a quick boy defeated a giant of a man.
^ 1999, The Queen's Two Bodies: The Double life of Elizabeth the First by Jeanne Murray Walker
The shelter was called the House of Ruth, which made me feel right at home since I know that Ruth was someone in the Old Book. We didn't read the Old Book, just the New, but still, I felt right at home with folks who were in it.
^ 2002, The Book of Fred by Abby Bardi
Corpus of Contemporary American English - 22 hits, some duplicates;
only 1 relevant hit, and it is the same as one that was found in the search of the Corpus of Historical American English (The Book of Fred)
NOW Corpus (News on the Web) - 77 hits; only 6 of them relevant, more or less
example from American political activist and documentary filmmaker, Michael Moore:
My own spiritual beliefs do not allow for capital punishment, and I was raised in the state (Michigan) that in the 1840s was the first government in the English-speaking world to outlaw it. So, I'm just not inclined that way. I don't believe in " an eye for an eye. " I know the old book said that, but I like its sequel better (a rare case in which the sequel -- like Godfather II, Star Trek II, Terminator II -- is better than the original). If you don't believe the way I believe (it's also the official position of the Catholic Church, for whatever that's worth these days), then that's your right, and I understand.
^ Some Final Thoughts on the Death of Osama bin Laden by Michael Moore
another example, from Nigeria
What God has joined let no man put asunder, the Old Book says.
^ in Najia News, 17 September 2017
Interestingly, of these uses listed here, only a handful unambiguously refer to the Old Testament. The rest could just as easily refer to the Bible (meaning the Old Testament and the New Testament together) as a whole.