Question about Californian shift and Canadian shift

For the speakers of those shifts, does the vowels of some "broad A" words like "father, Chicago" go together with "bother", i.e. they pronounce "father" as ["fQD@`] or even ["fOD@`]?

Yes, but this has nothing to do with the Canadian shift. Almost all North American accents are father-bother merged. Only parts of New England and maybe old-fashioned New York maintain this distinction. I don't think a rounded realisation of /A/ is as common in Canada as some textbooks indicate. I've heard it on automated phone messages, and it sounded weird to me, so it can't be the normal usage here in Ottawa.

Thank you, Josh Lalonde.

So the normal usage in Ottawa should be an unrounded [A], isn't it?

Yes, unrounded [A]. That's not to say that the rounded [Q] doesn't occur, but it's much less common. It does also occur due to allophonic variation in environments with /l/, as was discussed on the thread about 'bull' and 'dull'.

Sorry, in my first post, I meant to say that only parts of New England and New York preserve the distinction in North America. Outside of North America, all accents of English preserve the distinction, as far as I know.