Say I have a lacunary $q$ series $s(q)=\sum_{n=0}^{\infty} a_{n}q^{n}$ , and I have a quasiconformal transformation $\xi$ which preserves the boundary of the unit disk in $\mathbb{C}$ such that if $q=1$ then $\xi(q)=1$. Is there a method from Teichmüller theory that allows us to explicitly write down the coefficients $b_{n}$ of $s(\xi(q)) = \sum_{n=0}^{\infty} a_{n}\xi(q)^{n} = \sum_{n=0}^{\infty} b_{n}q^{n}$ given some explicit $\xi$?

The answer is no: pick $s(q)=q$ to be simply the identity, then you would obtain automatically a power series expansion for any quasiconformal $\xi$ preserving the circle, which can't be true since there are some $\xi$ that are not analytic near the origin... 

