In this paper, we introduce operators that are represented by upper triangular 2×2 block matrices whose entries satisfy some algebraic constraints. We call them Brownian-type operators of class Q, briefly operators of class Q. These operators emerged from the study of Brownian isometries performed by Agler and Stankus via detailed analysis of the time shift operator of the modified Brownian motion process. It turns out that the class Q is closely related to the Cauchy dual subnormality problem which asks whether the Cauchy dual of a completely hyperexpansive operator is subnormal. Since the class Q is closed under the operation of taking the Cauchy dual, the problem itself becomes a part of a more general question of investigating subnormality in this class. This issue, along with the analysis of nonstandard moment problems, covers a large part of the paper. Using the Taylor spectrum technique culminates in a full characterization of subnormal operators of class Q. As a consequence, we solve the Cauchy dual subnormality problem for expansive operators of class Q in the affirmative, showing that the original problem can surprisingly be extended to a class of operators that are far from being completely hyperexpansive. The Taylor spectrum approach turns out to be fruitful enough to allow us to characterize other classes of operators including m-isometries. We also study linear operator pencils associated with operators of class Q proving that the corresponding regions of subnormality are closed intervals with explicitly described endpoints.