would cost, does cost. "ask" is better than "see", but "see" is also used.
I usually say "would".
The strict indirect-speech style calls for:
I was calling to ask how much a haircut costs.
If you have to write that sentence in anything except fiction, I think it would be better to write it that way.
But since this is oral, and the salon is expecting a question, the order you've got is better:
I was just calling to ask, "how much would a haircut cost?"
I was just calling to ask, "how much does a haircut cost?"
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