  We illustrate the use of the theory of $qq$-characters by deriving the BPZ
and KZ-type equations for the partition functions of certain surface defects in
quiver ${\mathcal N}=2$ theories. We generate a surface defect in the linear
quiver theory by embedding it into a theory with additional node, with specific
masses of the fundamental hypermultiplets. We prove that the supersymmetric
partition function of this theory with $SU(2)^{r-3}$ gauge group verifies the
celebrated Belavin-Polyakov-Zamolodchikov equation of two dimensional Liouville
theory. We also study the $SU(N)$ theory with $2N$ fundamental hypermultiplets
and the theory with adjoint hypermultiplet. We show that the regular orbifold
defect in this theory solves the KZ-like equation of the WZW theory on a four
punctured sphere and one-punctured torus, respectively. In the companion paper
these equations will be mapped to the Knizhnik-Zamolodchikov equations
