In a math paper, is this considered redundant?

An issue with the first version is that it could indirectly suggests that not all matrices in that set are nonnegative. Then, somebody might wonder what "nonnegative" is supposed to mean. Could it mean the determinant is nonnegative or all eigenvalues or...?

This can create confusion and thus should be avoided. I agree though with the goal of putting emphasis on the fact, especially the first time it occurs. Consider this version or something along these lines instead:

...and thus there exists a matrix $\mathcal{Q} \in \mathcal{M}_N(\mathbb{R^+})$, i.e., a matrix with nonnegative entries, ....

Put differently, redundancy can be good, but it can make sense to make clear that information is redundant.

First of all, I'm not sure "nonnegative matrix" means, unambiguously, "matrix with nonnegative entries." But my bigger issue with

...and thus there exists a nonnegative matrix $\mathcal{Q} \in \mathcal{M}_N(\mathbb{R^+})$...

is that you may not need the notation at all. Sometimes programs that debug code will point out when you're declaring a variable but not using it for anything. It's not exactly an error, but it can complicate your writing.

If you're going to refer to the set of $N\times N$ matrices with nonnegative entries several times, name it at the beginning of a paragraph. Then use the shorthand notation later on. As in:

Let $\mathcal{M}_N(\mathbb{R^+})$ be the set of all $N\times N$ matrices with nonegative entries. ... yada yada yada ... and thus there exists $\mathcal{Q} \in \mathcal{M}_N(\mathbb{R^+})$ such that ...

This way, someone who missed what exactly $\mathcal{M}_N(\mathbb{R^+})$ was can scan backwards to the beginning of the paragraph to find it. Someone who remembers can just move on without stumbling over an inline declaration.

If you're not going to refer to the set often, there's no need to name it with notation. Just say

... and thus there exists an $N\times N$ matrix $\mathcal{Q}$ with nonnegative entries such that ...

Erdos wrote “the best notation is no notation.” Use it only if you need it.