In this paper, we derive several necessary conditions of turnpike property for generalized linear-quadratic (LQ) optimal control problem in an infinite dimensional setting. The turnpike property reflects the fact that over a sufficiently large time horizon, the optimal trajectories and optimal controls stay for most of the time close to a prescribed steady state of the system. We show that the turnpike property is strongly connected to certain structural-theoretical properties of the control system. We provide suitable conditions to characterize the turnpike propertyin terms of the detectability and stabilizability of the system. Later, we show the equivalence between the exponential turnpike property for generalized LQ and LQ optimal control problems.