"Before I start my reply, I want to point that, afaik, in Portuguese speaking countries, the most common way to delimit dialogues is to use only dashes. The quotation marks are reserved for thoughts and, sometimes, foreign words. I'm answering based on that criteria."
I handle that in my books as a normal dialogues.
– How could I be so complacent? - Alex said to himself.
In the phrase below, maybe it makes more sense to consider it as a thought, but I don't know to context.
"I will not give in." – Alex thought while climbing the mountain. – "I must continue on."
Normally I rule that, if a character expects an answer from himself, he is dialogging with himself, otherwise, he's just toughing about something... But that's only me. You should choose what works better for you.
I guess the answer on quotes, depends on how you think you should deal with the dialogue. If it's a dialogue, don't quote. If it's a thought, quote.