At the moment there does not appear to be a definitive history of the pub quiz, which is suprising as it appears to be a relatively recent phenomenon. Currently the wikipedia entry for pub quizzes has them as running since at least 1970 although the reference link for that is broken.
An article in The Times 15th December 1990 mentioned “The quiz has been part of pub life for many years. Tetley's, the Leeds brewery, has run a league for 22 years.” This suggests organised pub quiz leagues since 1968, but Tetley's is now part of Carlsberg and to be honest it is a tricky thing to google as each search term brings up lots of pubs with quiz leagues and serving tetleys.
The Times article also mentions "“Burns and Porter, Britain's biggest quiz organisation, runs 600 leagues for 16 breweries, supplying questions for 6,000 teams. After 14 years in the business, the company has just celebrated its two millionth quiz night”. Burns and Porter are much better documented and this is strong evidence that organized pub leagues were around since at least 1976.
I recently bought a copy of the "Sunday People Pub sports quiz book" (1979) which mentions that the newspaper's pub league has been running for 8 years, and was even televised! Sadly nothing on YouTube as yet. Apparently the idea for a national quiz league came from a Brian Madley, who as Midlands correspondent had noticed a craze for sports quizzes in pubs and clubs across Birmingham. This suggests that sports pub quizzes were popular in this area at around 1970/71.While looking for more information about the Sunday People pub quiz league I stumbled upon a pretty comprehensive history of the Bridgend Quiz League, which started in 1973 and is still going.Looking through regional newspaper archives I have found quite a few references to inter group quizzes as a social activity in the 1940s and earlier. These did not take place in pubs but look to be a clear pre-cursor to the pub quiz. At the moment I do not have complete access to the archive, something I need to sort out with my library!So far then I think I can say that I have strong evidence of a pub quiz league in Bridgend in 1973, and pretty strong evidence (just a bit vague at the moment) of a sport pub quiz league in Birmingham in the early 1970s and some evidence of a Tetley's league in 1968. I am going to try and track down some of the people involved in the early pub quiz leagues to see if they can help me in my quest to find the first pub quiz!