  Noether symmetry has been invoked to explore the forms of a couple of
coupling parameters and the potential appearing in a general scalar-tensor
theory of gravity in the background of Robertson-Walker space-time. Exact
solutions of Einstein's field equations in the familiar Brans-Dicke, Induced
gravity and a General non-minimally coupled scalar-tensor theories of gravity
have been found using the conserved current and the energy equation, after
being expressed in a set of new variables. Noticeably, the form of the scale
factors remains unaltered in all the three cases and represents cosmological
Lorentzian wormholes, analogous to the Euclidean ones. While classical
Euclidean wormholes requires an imaginary scalar field, the Lorentzian wormhole
do not, and the solutions satisfy the weak energy condition.
