  We construct cluster algebras the variables and coefficients of which satisfy
the discrete mKdV equation, the discrete Toda equation and other integrable
bilinear equations, several of which lead to q-discrete Painlev\'e equations.
These cluster algebras are obtained from quivers with an infinite number of
vertices or with the mutation-period property. We will also show that a
suitable transformation of quivers corresponds to a reduction of the difference
equation.
