The anecdote isn't true. There are reasons to approach this anecdote with skepticism:
Iroquois is not a single language but a whole family of languages (cf. John Lawler's comment or "Iroquoian languages" on Wikipedia).
Characters don't have to be right about everything they say. They may joke, prevaricate, or be ignorant.
There is a history of spurious anecdotes about indigenous languages or language use (cf. the idea that Eskimoan languages have many more words for snow (Wikipedia)), not to mention how common folk etymologies are in general.
I investigated three avenues to find evidence for the claim: early uses of cut no ice, Iroquoian vocabularies, and Iroquoian syntaxes. No search yielded evidence that the phrase comes from "Iroquois." That leads me to suspect (agreeing with this internet source) that this is all a joke by O'Brian.
The turn of phrase emerges in the late 19th century
See "ice, n." in the Oxford English Dictionary:
P5. colloquial (originally U.S.). to cut ice (with someone) and variants: to carry weight, have an influence or effect (upon); to impress. Chiefly in negative or other non-assertive contexts.
1894 J. A. Frye Fables Field & Staff 176 ‘Huh! w'at youse say cuts no ice wid me!’ says I, scornful. ‘It's clean nutty dat youse are.’
That is a late date for a phrase purportedly from "Iroquois" to cross over into English. This being a set phrase from Iroquois also wouldn't explain the productive variants like "cut any ice" (1896) or "haven't been cutting much ice" (1904).
Early usage would also make other claims about its origins. For instance, one 1894 article claims that the turn of phrase comes from Yale:
"Money cuts no ice," to use a Yale term, and a poor fellow is as popular as his friend across the hall who has an unlimited bank account. ("Yale Students' Haunts", Daily Inter Ocean, 10 December 1894.)
Another article refers generically to "the boys":
The fact that Attorney General Knowiton does not agree with City Solicitor Hopkins as to the power of the license commissioners to issue additional licenses [...] and that he has advised the police commissioners of Lowell that they have the right to issue such licenses, is interesting, but it "cuts no ice," as the boys say, in the local situation. ("The Liquor Licenses," Worcester Daily Spy, 20 July 1895.)
Whether any of those other etymological claims are true or not, I haven't found any usages that connect to a supposed indigenous origin.
The vocabulary doesn't support it
Even if we take a Swadesh list of Algonquian and Iroquoian languages (Wiktionary), it is apparent that the basic words we'd be looking for don't exist. For example, here is I in several languages:
There is nothing close to the words reported to mean "I am unimpressed": katno aiss' vizmi. The closest would be a Mohawk prefix like ka- for I, but that's shaky.
Similarly, one would be hard-pressed to find a translation for unimpressed in a dictionary or resource guide, let alone any similar word (like pleased or happy) that happens to look like the claimed phrase. While these resources aren't complete, they yield no evidence for words like the ones posed by O'Brian.
The syntax doesn't support it
This would take too long to answer completely, but take as an example Marianne Mithun's chapter "Noun and verb in Iroquoian languages: Multicategorisation from multiple criteria." Even in a brief read, one can see that verb words are formed through pronominal prefix, verb root, and aspect suffix. For instance, note these examples from Oneida illustrating the structure of several verbs:
That kind of structure doesn't admit the use of multiple word. Even if O'Brian had mistranscribed the form for the sake of fitting the character dialogue, it's hard to imagine words of this pattern in Oneida or similar structures.
Also, O'Brian's form doesn't feature the negative prepronominal prefix that may be expected with a translation of a negative like unimpressed, according to the Ontario Curriculum *Resource Guide* on Oneida, Cayuga, and Mohawk:
A verb with the negative prefix is always preceded by the negative particle yáh in Oneida, and iáh in Mohawk. In Cayuga, the negative particle te3› can sometimes be omitted.
So if ka- were somehow a first person pronominal prefix, we would expect another prefix before it.
For instance, negations in Oneida would look like this:
But O'Brian's example lacks that negative prefix. It's hard to imagine how one could take the syntactic rules presented in some of these languages and come out with something that looks like the claimed phrase. One can't prove a negative, but these rules give one nothing for positively affirming O'Brian's character's claim.
It's a joke
Finally we're left with the realization that we may have taken a novelist's joke much too seriously, dug into newspapers to find no early claims that it comes from an Iroquoian language, and dug into resources on several languages in a failed attempt to find parallels. So no, this is probably not a true anecdote.