I knew as soon as I mentioned it in passing someone would want to know immediately about the alleged ‘fart joke’ in the Bible. Ok, unless you’ve already googled it, here it is, but I should warn you that if you’re wondering why you haven’t seen it already it may because you need to know Hebrew, and Akkadian!
The story begins in Judges 1:11-15 (which is actually repeated from Joshua 15:15-19) where Caleb (one of the famous ‘two good spies’) offers a prize to whoever successfully conquers the city of Kiriath-sepher. The prize? His daughter Achsah as wife. As the story goes, it may have been better if Caleb consulted Achsah about this first, because she wan’t entirely happy about being given away to a hardened soldier as a trophy. The prize-winner was Othniel (later to become one of the famous judges in the book of Judges), Caleb’s nephew (and therefore Achsah’s cousin).
Achsah may have been a great prize for Othniel, but she wasn’t entirely satisfied with the deal herself and wanted something more, so she urges her cousin-now-husband to ask her father for some real estate to sweeten the deal. As the story goes, they come to Caleb and she dismounts from her donkey, prompting Caleb to ask “What do you wish?” (v.14). This is an odd question to ask in response to his daughter getting off her donkey! She asks for some land as a present, and receives it. (And it’s interesting, in passing, that it was Achsah, not Othniel, who did the asking, even though she’d tried to persuade her new hubby to put some pressure on the father-in-law).
The phrase וַתִּצְנַח מֵעַל הַחֲמוֹר “she dismounted from her donkey” includes a verb which occurs only here (in both accounts in Joshua 15:18 and Judges 1:14) and a couple chapters later in Judges 4:21, so translators are at best guessing with their translations. It’s an old problem and the earliest known translation (the Septuagint, an ancient Greek translation of the Hebrew Bible) says instead “she cried out from the donkey” (Joshua 15:18), or “she grumbled” (Judges 1:14) which are hardly the same as dismounting. The Septuagint translators may have misread ותצנח as ותצרח she cried,  or perhaps they read it right and another copyist read it wrong, and our Masoretic Text (MT) of the Hebrew Bible has descended from the copy with the mistake.
You’re probably becoming a tad impatient at this point and just want me to get to the fart joke! The eminent Biblical scholar Sir Godfrey Rolles Driver noticed a similarity with the Akkadian (Babylonian) word ṣanāḥu which he argued was the Akkadian term for farting, and so he translated the Hebrew as “she broke wind”, suggesting Achsah farted as a sign of her disgust and that was sufficiently clear to her father that he asked his otherwise odd question. One major English translation (the New English Bible) picked this up and included it in their translation. This may not be surprising, however, as G.R. Driver was the Convenor of the NEB Old Testament Committee! (and they dropped it in a later revision in favour of “she dismounted”).
As you may suspect, Driver’s translation received quite a bit of scholarly attention at the time and a fair bit of opposition, and hardly rates a mention in commentaries these days. It’s even been suggested that Driver was playing a joke of his own in slipping a fart joke into the Bible!
 As HALOT suggests. Koehler, Ludwig and Walter Baumgartner, The Hebrew and Aramaic Lexicon of the Old Testament. Study ed. Leiden: Brill, 2001, 1038.