今日は何をしようか。So is it a set rule that volitional form in plain (not using -ましょう) can use the question marker か?
According to "A Dictionary of Basic Japanese Grammar", page 166,
か can also be used in plain form.
これは本だ。This is a book.
これは本か。 Is this a book?
貴方は先生だ。You are a teacher.
貴方は先生か。 Are you a teacher?
In the examples above the だ drops and is replaced by か.
あの本は高い。That book is expensive.
あの本は高いか。 Is that book expensive?
私が話すか。 Do I speak?
In the examples above か is added at the end of the sentence.