Translation:Would you know where the nearest toilet is?
"Don't you (know)" in English is used for emphasis, and it often expresses at least a degree of surprise, frustration, or incredulity. If I accidentally walked into a broom closet when looking for the bathroom (restroom, toilet...) in my own home, someone might say to me, "Don't you know where the bathroom is in your own house?" (E.g., "Don't you know what day it is today? It's my birthday." "Don't you know how to do the simplest things?")
The "don't" / "doesn't" form can also be used for other questions in English to express some degree of doubt, in addition to the above (e.g., "Don't they live around here? I'm not sure."); and/or a sense that you had thought something was true ("Don't you have a cat? I thought I saw one in your living room last weekend.")
This isn't a comprehensive list; it just suggests some examples.
I have the sense that negative questions in Czech can simply have a more polite connotation than direct, positive questions. (E.g., as tylerskarz suggested, "You wouldn't happen to know...", or something similar.)
Jmm... Actually the point is that I was understanding the phrase as a doubt phrase. But, it seems that, for czech, it has a polite meaning. Anyway, it would be great to know if czech negative questions have also that "doubt" connotation.
Any czech could solve this doubt? :)
Okay, for that case maybe it is just me. But I do think the preferred translation should be "Do you know" https://books.google.com/ngrams/graph?content=Would+you+know+where%2CDo+you+know+where&year_start=1500&year_end=2008&corpus=15&smoothing=3&share=&direct_url=t1%3B%2CWould%20you%20know%20where%3B%2Cc0%3B.t1%3B%2CDo%20you%20know%20where%3B%2Cc0
No, you can ask anyone like this. It is completely fine this way if you are on the T side of the https://en.wikipedia.org/wiki/T%E2%80%93V_distinction with this person (tykání).
"Don't you know" should be accepted because it's the literal translation and we're not here to guess what the intention was. Obviously Slavic people including myself would use the negative to ask if someone knows by any chance about that bloody toilet but refusing to accept the literal translation is just silly. How else would you say "Don't you know where the nearest toilet is?" in Czech?
I would personally translate "Don't you know where the nearest toilet is?" as "Copak nevíš, kde je nejbližší záchod?", which expresses the surprised or incredulous tone of the English negative question equally well. Another possibility would be "Ty snad nevíš...?" or at the very least "Ty nevíš...?"