We introduce quantized Chebyshev polynomials as deformations of generalized
Chebyshev polynomials previously introduced by the author in the context of
acyclic coefficient-free cluster algebras. We prove that these quantized
polynomials arise in cluster algebras with principal coefficients associated to
acyclic quivers of infinite representation types and equioriented Dynkin
quivers of type $\mathbb A$. We also study their interactions with bases and
especially canonically positive bases in affine cluster algebras.