A Hilbert Space operator T is of class Q if $T^2{ast}T^2-2T{ast}T + I$ is nonnegative. Every paranormal operator is of class Q, but class-Q operators are not necessarily normaloid. It is shown that if a class-Q contraction T has no nontrivial invariant subspace, then it is a proper contraction. Moreover, the nonnegative operator Q = $T^2{ast}T^2-2T{ast}T + I$ also is a proper contraction.