Is there a combinatorially defined, nonempty effectively closed set $Q\subseteq 2^\omega$ such that all members of $Q$ are incomputable?
Combinatorially defined means that the definition of $Q$ does not involve any logic or computability theoretic notion. This excludes those effectively closed sets defined by PA degree, 1-randomness etc. Effectively closed set means the set $\{\rho\in 2^{<\omega}:[\rho]\cap Q=\emptyset\}$ is comuptably enumerable.
The answer is yes.
Let me first describe a general method. Fix any c.e. computably inseparable pair $A$ and $B$. These are computably enumerable sets having no computable separation. There are diverse examples of such pairs of sets. Let $Q$ be the set of sequences $s$ separating $A$ and $B$, that is, for which $s(n)=1$ whenever $n\in A$ and $s(n)=0$ whenever $n\in B$. By assumption, there are no computable elements of $Q$. The set is effectively closed, because we can recognize nonmembers of $Q$ in finite time, simply by waiting for the elements of $A$ and $B$ to emerge and observing whether the membership requirements for $Q$ were followed or not.
Next, let me complete the answer by mentioning that there are numerous examples of computably inseparable pairs, including examples that I would think of as combinatorial.
Let $A$ be the set of Turing machine programs $p$ that halt on empty input with output $0$, and let $B$ be the set of programs that do so with output $1$.
Let $A$ be the set of theorems of PA and $B$ the set of negations of theorems of PA.
Let $A$ be the set of finite Game-of-Life positions that lead eventually to square $1$ turning on before square $2$ (using two fixed squares that makes this example work) and $B$ the set of positions in which $2$ turns on before $1$. This is computably inseparable, because Game-of-Life can simulate Turing machines, and we can pick the squares so that square $1$ corresponds to accept and square $2$ corresponds to reject.
For a particular finite set $\Gamma$ of rational polygonal tiles (or Wang tiles, if you prefer), let $A$ be the set of finite additional possible tiles of a particular form, which admit no tiling of the plane when added to $\Gamma$, and let $B$ be the set of such augmented tile sets of another particular form that admit no tiling when added to $\Gamma$. Since the tiling problem is Turing complete, I can make an example of this form, where the additional tiles of $A$ correspond to halting with output $1$ and those of $B$ correspond to halting with output $0$, but there is no need to talk about Turing computation to describe the actual tiles.
Let $A$ be the set of Post correspondence problems (of a certain form) that admit a solution ending with the symbol $0$ and $B$ the set of such problems admitting a solution ending with the symbol $1$. Since the Post correspondence problem can simulate Turing computation, this example can be created from the first example above.
I think one could create many more examples of a combinatorial nature, simply using the fact that combinatorial questions are often Turing complete and can encode Turing computations.
