Curly quotes are separate unicode characters from "normal" straight quotes. I see the curly ones just fine with Sigil. The font-substitution engine should pick them up with no problems if the default font doesn't have them.
The default font sometimes makes it a bit hard to tell the curly ones from the normal ones, but the difference is there. The Code View uses a monospaced font and should always show the difference clearly.
It's possible that curly quotes are not being rendered properly on your machines. Try loading Zelda's
Three men in a boat epub in Sigil. That uses curly quotes that show up just fine on my machine. If they don't show up for you, then that's a problem.
But when you type in quotes, Sigil will not insert curly ones for you automatically. I said everything I'm going to say on the topic in response to
issue 275, and you can look there for the explanations why it unfortunately works the way it does.