There's a formulation of the liar sentence that goes, "I am lying" (kind of the reason we call it the liar sentence in the first place, I suppose, except historically relativized to Cretans/Epimenides). So if the alethic nihilist's number-one argument for the incoherence of the phrase "is true or false" is that the concepts of "true" and "false" result in a contradiction via the liar paradox, why wouldn't the alethic nihilist think that, "I am lying," in generating the liar paradox, then reveals the concept of "lying" and "honesty" to also be deficient?
I say that not to answer in place of a genuine alethic nihilist, however. For they might not think to take, "I am lying," to be genuinely paradoxical so much as pragmatically empty (in terms of the semantics/pragmatics distinction in the philosophy of language). The use of the "I" here might make the I-liar sentence into something not abstract enough for the alethic nihilist to "care" about. Or they could agree that, when, "I am lying," is stipulatively equivalent in purpose to, "This sentence is false," this is when the word "lying" loses its final value; but this is not to disallow a stipulative other definition of "lying" in moral terms that the alethic nihilist accepts on the side.
Per my comment, I think that the alethic nihilist would be hard pressed to uphold the substance of such a claim, though. Not only does the alethic nihilist apparently have to deny the correspondence, coherence, etc. theories of truth, but even axiomatic theories of truth are in tension with their thematique, sources of potential discomfort for them in light of their "worldview." Yet stipulative definition is a powerful enough tool in the realm of axiomatics (not as powerful as we'd like, sadly or not) that it seems like a specifically arete-theoretic and axiomatic theory of truth, in which the inference to the liar paradox is blocked in some required way, should not be too much trouble to construct in abstracto. The alethic nihilist might gain a world but lose their alethic soul (to put it overdramatically), because they will be denying that we have the right to define "true" in a stable way based on an aretaic sense. (And so for that, even that alone, they might be inspired to admit that at least the aretaic theory of truth is allowable for a good enough reason, while concluding that every other (family of) theory of truth is to be denied, almost as strong a conclusion as if they denied any and all uses of the word "true" whatsoever.)
Historical addends: I should like to add that the word "truth" reflects the word "troth" as in "betrothed," so in English the affinity of the concept of truth with the concept of fidelity (a type of specifically romantic virtue) is not hard to identify. Moreover, a similar linguistic situation arises in Latin, where veritas = truth resonates with the word virtus itself (although I am not aware that the similarity is as close in spirit as that is for "truth"/"troth"). And though nowadays it is rather odd to read the New Testament and see the alleged Son of God ordering people not to make vows but to let their yes's be yes's and their no's no's, "For anything beyond this is of the Evil One" (the very Father of Lies, the stories say!), sense can be made of this if we think of the still-occurrent practice of swearing to tell the truth in court: the alleged champion of Heaven and Earth did not mean to forbid promise-making in general but the peculiar form of promise-making that violates the holy spirit of the veritas/virtus connection (not his words, but their spirit still).
Finally (for present purposes), Thomas Aquinas sometimes wrote of the "truth of things," as if objects, not just assertions or possible thoughts (Leibnizian propositions), can be true (or false). I have often debated whether holding that, "This sentence is false," is asserting object-falsity, not sentence-falsity, of itself, would resolve the liar paradox, although I can easily envisage a revenge paradox for, "This sentence is sentence-false" (unless that's too "insane" a sentence to allow except philosophically).