Caveat: My Twine (Twine 2.1.3, Harlowe 2.0.1) doesn't do that. That said, a few comments:
For quoted text, you should be using the <q>...</q> HTML tags wherever possible. They can be nested, they then make sure your quotes are "balanced", and they automatically create the correct opening/closing quotes when supplied with the correct CSS. For example, this simple CSS gets you the correct English quotes.
q {
quotes: "“" "”" "‘" "’";
}
Also, you shouldn't be using an apostrophe, you should be using the right single quotation mark in English text. Overall, the line should be written like this:
<q>I don’t know. I haven’t tried that.</q>
The quote tag goes a bit further than that, you can use it to have different quotes for different languages, or even style the quotes passages differently depending on who's speaking by using custom attributes. For example, if you wish to use Japanese quotes when lang="ja" and the English ones otherwise, it's one extra CSS rule:
q {
quotes: "“" "”" "‘" "’";
}
q:lang(ja) {
quotes: '「' '」' '『' '』'
}
And then just add the correct "lang" attribute:
<q>As they say… <q lang="ja">ありがとう</q>.</q>
Or if you want a whole paragraph to be in a different language:
<p lang="ja"><q>大丈夫ですか?</q></p>