Writing quotation marks on iOS with a braille display

Did you try just dot 4?
Do that for open and closed quotations. I think everything you tried didn't work because you were not using computer braille symbols. And in cumputer braille, open & closed quotations are dot 4. Hope this helps!

For me dot three is a single quote aka apostrophe and dot five is normal double quote.

Hello: I also do use UEB as my primary Braille input table and have begun using it more recently when doing my own coding. In HTML, when writing up Web pages it is quite easy to do it.
I usually get around these things, like when inserting an HTML attribute after the equals (=) sign, by using a standard Braille close quotation mark, dots 3,4,5. An example of this would be such as: <a href="http://apple.com/"&gt;
For this I used the dots 3,4,5 before and after to surround the link tag URL. Then you would get on with the text as normal. The reason for this is that in UEB, a left quotation mark of dots 4,5, then dots 2,3,6 is treated absolutely even in computer translation. It has to do with how it looks like in Print which, with HTML coding, wouldn't work either. As it happens, I am using VoiceOver with Braille Screen input on my phone, and “yes, it even works that way” with the previous example and it translates it well.
If you use Computer Braille, at least the USA table of Computer Braille, the quote is dot 5.
Hope this helps.

Yes, it is dots 3-5-6. I'm just thinking that Braille has become too much of a first and second nature to me that I don't really think about my dot combinations... Smile out loud...
Also, my code sample keeps not working; the site might be trying to render it as actual code.
But yes, try doing dots 3-5-6 for the quotation mark, as dots 2-3-6 will do a question mark in UEB.