This question is frequently asked, usually in relation to the Riemann hypothesis or the consistency of set theory. I suppose this is as good a place as any to collect references to explicit arithmetizations of initially non-Diophantine problems, as (systems of) Diophantine equations.

In one of his earlier papers, Matijasevich gives an example of a Diophantine equation expressing a number-theoretic statement, different from the ones on the path to Hilbert's 10th problem. The ones on the path were his breakthrough "$m = F_{2n}$" and the universal Diophantine predicate, "machine $m$ halts in time $t$ on input $n$". The illustrative example was something like "is a prime number", not anything as complicated as consistency of ZFC. The universal equation can be specialized by a choice of $m$ and $n$ to an equation expressing inconsistency of ZFC, but actually writing down specific values accomplishing that translation would be a formidable task. (ADDED: the paper is online at Link )

J.P. Jones has several papers, some with Matijasevich, providing Diophantine encodings of other number theoretic statements. Website: http://math.ucalgary.ca/~jpjones/papers.htm

The long paper by Davis, Robinson and Matijasevich gives Diophantine representations of the Riemann Hypothesis and of "$p$ is a prime number" from which one can write down the Goldbach conjecture, or near-equivalents of the Twin Primes conjecture. Most of it is online at: http://books.google.com/books?id=4lT3M6F745sC&pg=PA323

I don't know if (in)consistency of ZFC has been displayed in print as a Diophantine equation.
There may be computer programs available to perform the transformations from ZFC parser to Turing machine to Diophantine representation, and if so, their possibly very large output would answer the question. If you are willing to allow exponentiation as a primitive, as Goedel did in his paper, Gregory Chaitin made software available to construct a huge exponential Diophantine equation, similar to the one printed in his book, whose solution set is algorithmically random when projected onto one of its variables. The Matijasevich-Robinson encoding of $a=b^c$ could then be applied, to produce an ordinary, but even larger, Diophantine statement.

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