One explicit connection is to look in non-describing characteristic, i.e. over an algebraically closed field $k$ of characteristic $p$ not dividing q, so $q$ can be simultaneously a root of unity and a prime power.

Let's say we are in type $A$, then the Iwahori-Hecke algebra $\mathcal{H}_q(n)$ and the $q$-Schur algebra $S(q,n)$ over $k$ arise as endomorphism algebras of certain unipotent representations of $GL_n(\mathbb{F}_q)$ and control many aspects of the unipotent representations, in fact there is an equivalence of categories between $S(q,n)$-modules and unipotent representations (for $\mathcal{H}_q(n)$-modules there are biadjoint functors, but not equivalences in general).

For example we can see that if the order of $q$ (mod p) is bigger than $n$ then the order of $GL_n(\mathbb{F}_q)$ is prime to $p$ and hence every representation is completely reducible. This corresponds to the fact in characteristic zero that the algebras $\mathcal{H}_q(n)$ and $S(q,n)$ fail to be semisimple exactly when $q$ is a root of unity of order less than $n.$

One can relate the cases of positive characteristic and characteristic zero using model theory. In particular if you let the characteristic $p$ tend to infinity while always choosing $q$ to be a primitive $d$th root of unity then the representation theory of $S(q,n)$ (or $\mathcal{H}_q(n)$) in characteristic $p$ "converges" (in a model theoretic sense) to that of $S(\zeta,n)$ in characteristic zero (where $\zeta$ is a primitive $d$th root of unity).

Hence you can prove things about Schur algebras and Iwahori-Hecke algebras at roots of unity in characteristic zero (and therefor about quantum groups) by looking at unipotent representations of $GL_n(\mathbb{F}_q)$ in non-describing characteristic. For example it's not too hard to turn the above paragraph into an actual proof of for which values of $q$ these algebras are semisimple in characteristic zero.

One explicit connection is to look in non-describing characteristic, i.e. over an algebraically closed field $k$ of characteristic $p$ not dividing q, so $q$ can be simultaneously a root of unity and a prime power.

Let's say we are in type $A$, then the Iwahori-Hecke algebra $\mathcal{H}_q(n)$ and the $q$-Schur algebra $S(q,n)$ over $k$ arise as endomorphism algebras of certain unipotent representations of $GL_n(\mathbb{F}_q)$ and control many aspects of the unipotent representations, in fact there is an equivalence of categories between $S(q,n)$-modules and unipotent representations (for $\mathcal{H}_q(n)$-modules there are biadjoint functors, but not equivalences in general).

For example we can see that if the order of $q$ (mod p) is bigger than $n$ then the order of $GL_n(\mathbb{F}_q)$ is prime to $p$ and hence every representation is completely reducible. This corresponds to the fact in characteristic zero that the algebras $\mathcal{H}_q(n)$ and $S(q,n)$ fail to be semisimple exactly when $q$ is a root of unity of order less than $n.$

One can relate the cases of positive characteristic and characteristic zero using model theory. In particular if you let the characteristic $p$ tend to infinity while always choosing $q$ to be a primitive $d$th root of unity then the representation theory of $S(q,n)$ (or $\mathcal{H}_q(n)$) in characteristic $p$ "converges" (in a model theoretic sense) to that of $S(\zeta,n)$ in characteristic zero (where $\zeta$ is a primitive $d$th root of unity).

Hence you can prove things about Schur algebras and Iwahori-Hecke algebras at roots of unity in characteristic zero (and therefore about quantum groups) by looking at unipotent representations of $GL_n(\mathbb{F}_q)$ in non-describing characteristic. For example it's not too hard to turn the above paragraph into an actual proof of for which values of $q$ these algebras are semisimple in characteristic zero.