A cluster algebra, which was introduced by Fomin and Zelevinsky, is a commutative algebra with a family of distinguished generators (the cluster variables) grouped into overlapping subsets (the clusters) which are constructed by mutations. A quiver Grassmannian is a projective variety parametrizing subrepresentations of a quiver representation with a given dimension vector. After introducing how cluster algebras are related to the Euler characteristics of quiver Grassmannians, we give explicit expressions for the Euler characteristics in the rank 2 case.