Ah--an intelligent kid's question has worked its way into BirdChat channels. Since I worked on nighthawk digestion during my ill-fated Ph.D. research (the world already has one-too-many Dr. Laura's anyway, right?) I figured I better tackle this question. So I plunged in, noting that nighthawks have extremely odoriferous droppings because mixed with the fecal matter and urates are also caecal contents, and the caeca are filled with anaerobic bacteria that, as far as I've been able to determine, break down some of the chitin in their insect diet. While studying them, I spent many, many hours in extremely close quarters with nighthawks, paying an inordinate amount of attention to sounds and smells, and never once detected anything resembling a fart.
That said, mammalian intestines are very long, digestion slow, and fecal contents can remain in the intestines for quite a while. Gases erupting from any point in mammalian intestines can build up before they reach the end, and are going to pass through a bunch of malodorous stuff along the way, picking up additional odors. Birds get rid of any gases as quickly as they do the other material in their guts, and so there really isn't time for huge buildups of the kind that 10-year-old children (of both sexes, in my experience both as a former kid and as a former teacher) delight in and squeamish, oh-so-proper adults (again, of both sexes) recoil from.
Then someone wondered what it is in human nature that is attracted to such a question. To which I simply had to jump in again. What is it that attracts us to any question? We humans have five senses, and although our sense of smell isn't as keen as that of a golden retriever, anyone who's spent any time in proximity to a golden retriever in the evening (well, at least near MY dear, departed golden retriever Bunter), cannot help but have noticed the phenomenon we call farting. We upright humans may not produce quite as powerful a smell, but our own farts do at least occasionally make up for in sound production what they lack (only marginally) in odoriferousness. So only a deaf person who also lacked an ability to detect odors could possibly have gone through life never experiencing the phenomenon of farting, and only a human without an operational mind could possibly have gone through life never once wondering what the heck farts are all about.
I suspect my golden retriever didn't think much about the phenomenon, but she also didn't care who Ernie Banks was, or wonder if he rejoiced, in his retirement, at having played for the Cubs long enough (just barely) to have been able to play against a flock of Blue Jays. My golden retriever probably never wondered about the stars in the sky or whether there is a God or why cats bury their feces and dogs don't or why manufacturers don't make a 20x fixed-focus eyepiece for digiscoping. She just knew that anytime anyone acted upset about anything, it was somehow her fault, so when anyone said "EWWWWWWWW!" in the evening, she guiltily slunk over and nestled her head in their lap to make them feel better. Which of course was counterproductive, but she never thought about that, either. We mammals pretty much all fart, and presumably most of the humans among us think, and though few among us will ever produce a fart that approaches the sheer smelliness of a golden retriever's, few of us will ever be as kind as one, either.
It's when we recoil from asking natural questions that we are least human and most like animals.
But the short answer is an almost definite no, birds do not fart. Farts are, by definition, noticeable eruptions of significant volumes of intestinal gas. Avian intestines are short and evacuate wastes frequently. Any gases produced in digestion leak out as fast as they're produced, so there isn't the opportunity for build-up that leads to those explosive releases we cheerfully or disgustedly call farts.