If seeking for exact answer, then there is no way. Notepad++ does not have this functionality. Perhaps there's some less known Notepad++ plugin, but I really don't know about it.
But this is how I am getting access to special characters: using the following AutoHotKey macros you can map standard keyboard sequences into producing quotes or other character or sequences. Basically, it does comfortable replacement of one sequence for another. This isn't auto-recognition for switching quotes, but such a thing would require writing new Notepad++ plugin at least.
The AutoHotKey macros for quotes:
:O:;,::‘ :O:;'::’ :O:;,,::“ :O:;''::” Based on the above, type ;,,(followed by space or comma or newline to start replacing) and you will get “.
Macro example apart from creating quotes:
::;td::<td></td> By the above example you can produce table cell <td></td> instantly by typing ;td. So macros are universal, not only for quotes. I'm using semicolon as first character of sequences, because typing plain td would obviosly have undersired effects when typing plain text. But you can choose anything else, e.g. ` (backtick).
Advantage: macros you defined now work across all your applications (unless you restrict them).
So the format is
::type-this::replacement :O:type-this::replacement after typing type-this, you need to type space, comma, newline, etc. With first form, those are preserved, with second mentioned form, that character is consumed.
2017-02 Update:
I realized that AHK Macro can be written to rotate between quotes.
Example:
" → ” → “ –(restart)→ " → ” → ...
Let me know if you are still interested.