According to Daniel Cassidy, in his lying piece of trash, How The Irish Invented Slang, the word beak, an old English slang term for a constable or a judge or a schoolmaster, comes from the Irish beachtaí or beachtaire.
According to Cassidy’s book:
Beak, n., a judge, a magistrate.
Beachtaí, beachtaire, n., a critic; a correcting, captious judgmental person; fig. a judge. Beacht, al. beachd (Gaelic), n., judgment, opinion.
What’s wrong with Cassidy’s argument? Well, the main thing is the pronunciation. Most people reading Cassidy’s book would probably assume that beachtaí is pronounced as beek-tay or beek-tee. Cassidy probably thought the same, because his knowledge of Irish was practically nil. In fact, beachtaí is pronounced bach-tee, with the ch more or less an h sound or the ch of Scottish loch or the j of Baja California. It sounds nothing like beak. As for the meaning, a beachtaí (or its variant beachtaire) is a quibbler, a hair-splitter. It does not mean a judge. As we’ve pointed out before, where the letters fig. are used in Cassidy’s book, they stand for figment of Cassidy’s imagination, not for figuratively as they do in most books. O’Dónaill’s dictionary defines it as “Critical, captious, person.”
It is true that beachd can be a noun meaning judgement in Scottish Gaelic but Scottish Gaelic is a different language entirely. This meaning isn’t found in Irish.
So where does beak come from? The simple answer is, we don’t know. You can find a few suggested origins here: http://www.businessballs.com/clichesorigins.htm