Is “of” indispensable to the sentence, “’Is it true,’ asked Josef of his mother ‘that he is not our brother?’”

When the two words are in the order Josef asked, then no preposition is required:

"Is it true," Josef asked his mother, "that he is not our brother?"

Moreover, if we don't need to specify who Josef is asking, then no preposition is required:

"Is it true," asked Josef, "that he is not our brother?"

However, when we use the order asked Josef, and we want to specify who Josef is asking, then we need some preposition:

"Is it true," asked Josef of his mother, "that he is not our brother?"
"Is it true," asked Josef to his mother, "that he is not our brother?"
"Is it true," asked Josef from his mother, "that he is not our brother?"

Why of, instead of to or from? That's hard to explain. I just checked Macmillan (5 definitions of ask, 21 definitions of of) and Collins (25 definitions of of, 10 definitions of ask), and I could find no definitive entry that made the answer to your question obvious. The best answer I can provide from those sources is Meaning #2 of ask from Collins:

ask [trans. verb] to put a question to (a person); inquire of

but even that doesn't make it evident why we need a preposition when the word ordering is Josef asked, but not if the word ordering is asked Josef, or why of would be the preposition of choice.

Suffice to say that you shouldn't feel unenlightened if you feel confused by the issue. It's not something that's easily discerned by studying dictionary definitions.

Is of indispensable in the sentence - “‘Is it true,’ asked Josef of his mother, ‘that he is not our brother?”

Yes, unless the sentence is reworded to a form like “‘Is it true,’ Josef asked his mother...”.

Does it make sense [to write] “‘Is it true,’ asked Josef (to) his mother,’” by omitting ‘of’?

No. One could (and should) omit of in the following: “‘Is it true,’ Josef asked of his mother...”