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Then only I will say: "You are right".

--- Is it OK as it is, or should it rather be "Then only will I say: "You are right".

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Which word do you intend "only" to modify? Is it "I"? Or could it be "then"?

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I think "then".

In that case, you want:

Only then will I say "You are right".

And if "only" modified "I" then the original sentence would be correct?

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... or "Only then will I say (that) you are right", of course.

Newguest

And if "only" modified "I" then the original sentence would be correct?

Yes, that's right.