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](https://mathoverflow.net/a/273121/1946), 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](http://jdh.hamkins.org/km-implies-conzfc/), 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: <cite authors="Hamkins, Joel David; Leahy, Cole">_Hamkins, Joel David; Leahy, Cole_, [**Algebraicity and implicit definability in set theory**](http://jdh.hamkins.org/algebraicity-and-implicit-definability/), Notre Dame J. Formal Logic, vol. 57, iss. 3, pp. 431-439, 2016. [doi:10.1215/00294527-3542326](http://dx.doi.org/10.1215/00294527-3542326),  [ZBL06621300](https://zbmath.org/?q=an:06621300).</cite>)