This is, sadly, not as cool as it sounds.
There are two facts here:
- "Large cardinals imply consistency."
Large cardinal axioms have high consistency strength. For example, the existence of an inaccessible implies the consistency of ZFC, the existence of a weakly compact implies the consistency of ZFC + "there are a proper class of inaccessibles," etc. While at higher levels these arguments can be quite complicated, the lower-level instances are quite simple: e.g. to show that ZFC + "there is an inaccessible" proves Con(ZFC), you just have to argue that $V_\kappa$ satisfies ZFC whenever $\kappa$ is inaccessible.
- "Consistency can be represented by a polynomial."
This is an improvement of Godel's result that "consistency can be represented by arithmetic;" while very cool, it should be understood as not too unexpected, given Godel's theorem. Precisely, if $T$ is a recursively axiomatizable theory then there is a (multivariable) Diophantine equation $e$ such that a very weak theory - say, PA - proves that $T$ is consistent iff $e$ has no solution. (This is closely related to the MRDP theorem, which says that arbitrary c.e. sets can be represented by polynomials - see e.g. these slides.)
So putting these together, we get for example (assuming that ZFC is consistent) that there is a Diophantine equation $e$ which ZFC + "There is an inaccessible cardinal" proves has no solution, but which ZFC alone does not prove has no solution.
This is really about the consistency strength rather than the actual large cardinal. The statement that a given Diophantine equation has a solution is quite simple, and in particular is absolute between $\omega$-models of pretty much the weakest set theory you can imagine. So we can never have an unsolvability statement imply the actual existence of a large cardinal.