  In this paper, we generalize the Markov triples in two different directions.
One is generalization in direction of using the $q$-deformation of rational
number introduced by \cite{MO} in connection with cluster algebras, quantum
topology and analytic number theory. The other is direction using castling
transforms of prehomogeneous vector spaces \cite{SaKi} which plays an important
role in the study of representation theory and automorphic function. In
addition, the present paper gives a relationship between the two
generalizations. This may provide some kind of bridging between different
fields.
