Assumptions/definitions: the Gödel sentence is informally equivalent to, "This sentence can't be proved in system X," where X is appropriately specified. Since that sentence can itself be created inside such a system, then, the system can be seen as giving rise to self-unprovable sentences, such that the system is not able to completely prove itself (when taken as a conjunction of all the sentences that can be given in and from the system).
Now, erotetic logic is centered on questions, but, "This question cannot be answered in system Y," is ill-stated (since it is not a question). The concept of (abstract) problems is quite erotetic in character, but we often use the word "problem" in such a way as indicates a factive aspect/component to what problems are. So we assume (or hope) that, "This problem is unsolvable in system Y," is either well-stated or ill-stated in a different way. If it is still ill-stated, my first suspect in the case would be representing such a sentence as a problem: how is it a problem-to-solve (by some process)? Can a sentence just declare of itself that it is problematic "by stipulative definition"?
If, "This problem is unsolvable in system Y," goes through, and retains a deep erotetic flavor while existing also in substantively factive terms, then is it possible to state the first incompleteness theorem in these same terms? I'd be hard pressed to claim that I knew what erotetic Gödel codes would look like (if you will), but as an outsider looking in, I wouldn't expect that the coding would be prohibitively difficult.
Upshot? The Gödel sentence and the attendant incompleteness issues have been noted to be similar to the knower-paradox sentence and Fitch's paradox of knowability. But if axioms are knowable, then knowledge goes beyond proof, and at any rate, "to solve a problem" seems expansively inclusive over "to prove a point" in that proofs are solutions to problems (of validity/inference) but there are problems with non-proof solutions (or with both provable and unprovable, but either way knowable, solutions, perhaps). So instead of allowing for a newfangled image of the incompleteness problem, does the unsolvable sentence emerge just as a shadow of the knower-paradox sentence?
EDIT: to me, the unsolvable sentence seems potentially true. If it can be a problem by defining itself that way, it might be yet given to define itself so obtusely that it is, however, unsolvable (in that it lacks the "word problem details," so to say, that we'd need to solve it). So it is an unsolvable problem period, arguably, not just in some system or another. But this would favor the analogy with the unknown sentence rather than the unprovable one, maybe.