Yes, the assertion, "there is a truth predicate" is expressible in the language of second-order set theory. It is expressible in the second-order language of set theory, that is, the language of GBC or KM, rather than the first-order language of set theory. I gave the definition of what it means to say that a class $T$ is a truth predicate in my answer to your other question, to which you linked.
It follows of course that the non-existence of such a predicate is also expressible. This is a single $\Pi^1_1$ assertion in the second-order language of set theory.
The theory GBC+"there is no truth predicate" is equiconsistent with ZFC, since clearly the consistency of this theory implies the consistency of ZFC, and conversely, if there is a model of ZFC, then there is a model of GBC having only definable classes, and this model has no truth predicate. So the assertion that there is no truth has no large-cardinal consistency strength.
In contrast, the assertion that there is a truth predicate does transcend ZFC in consistency strength, since it implies Con(ZFC) and Con(Con(ZFC)) and much more, as I explain in my blog post, to which you linked.
Meanwhile, the truth predicate, when it exists, although it is not first-order definable, is nevertheless first-order implicitly definable (and hence first-order algebraic), since when it exists it is the unique class with that first-order property. (See more about this concept of implicit definability and algebraicity in my paper: Hamkins, Joel David; Leahy, Cole, Algebraicity and implicit definability in set theory, Notre Dame J. Formal Logic, vol. 57, iss. 3, pp. 431-439, 2016. doi:10.1215/00294527-3542326, ZBL06621300.)