I’m a little curious if this is by design or accident:
I’ve noticed that whenever I type a " [quotation] mark, even when set to italics, I always get the straight version, and thus to achieve the second, curved/italic mark I have to use the font character viewer. Is this intentional and is there a way around it? I prefer the curly italicized version. (I should mention this is in a subtitle in a text box, not a lyric.)
It’s wierd that Lyrics does it properly and text frames don’t. It might be a Qt thing.
If you’re on a Mac, the curly quotes are Alt [ and Shift Alt [, with ] being for single quotes. (On a US/UK keyboard: other keyboards may differ.)
Yeah this is a long-standing limitation (if that’s the right word). Text frames or text boxes don’t convert to curly quotes (or apostrophes).
On Windows, it’s Alt-0147 and Alt-0148 for quotes, and Alt-0146 for apostrophe. Those keystrokes are good friends of mine. 
Well, I’m glad I’m not crazy. I don’t use subtitles all that often, but I’ve noticed it once or twice before and assumed it was a limitation of my font choice. Then out of curiosity I tried the character viewer and they popped right in and I realized something odd was going on, especially since they populate in the lyrics.
PopChar is my friend. 