  Gravitational model of non-minimally coupled Brans Dicke (BD) scalar field
$\phi$ with dynamical unit time-like four vector field is used to study flat
Robertson Walker (RW) cosmology in the presence of variable cosmological
parameter $V(\phi)=\Lambda\phi.$ Aim of the paper is to seek cosmological
models which exhibit metric signature transition. The problem is studied in
both classical and quantum cosmological approach with large values of BD
parameter $\omega>>1$. Scale factor of RW metric is obtained as
$R(t)=6\sqrt{\frac{3}{\Lambda}}\cosh\big(\frac{t}{4}\sqrt{\frac{\Lambda}{3}}\big)$
which describes nonsingular inflationary universe in Lorentzian signature
sector. Euclidean signature sector of our solution describes a re-collapsing
universe and is obtained from analytic continuation of the Lorentzian sector by
exchanging $t\to it$ as
$R(t)=6\sqrt{\frac{3}{\Lambda}}\cos\big(\frac{t}{4}\sqrt{\frac{\Lambda}{3}}\big).$
  Dynamical vector field together with the BD scalar field are treated as fluid
with time dependent barotropic index. They have regular (dark) matter dominance
in the Euclidean (Lorentzian) sector. We solved Wheeler De
  Witt (WD) quantum wave equation of the cosmological system. Assuming a
discrete non-zero ADM mass $M_j=4\sqrt{2}(2j+1)\sqrt{\frac{\Lambda}{3}}$ with
$j=0,1,2,\cdots$, we obtained solutions of the WD equation as simple harmonic
quantum Oscillator eigen functionals described by Hermite polynomials. Absolute
values of these eigen functionals have nonzero values on the hypersurface
$R=6\sqrt{\frac{3}{\Lambda}}$ in which metric field has signature degeneracy.
Our eigen functionals describe nonzero probability of the space time with
Lorentzian (Euclidean) signature for $R>6\sqrt{\frac{3}{\Lambda}}$
($R<6\sqrt{\frac{3}{\Lambda}}$). Maximal probability corresponds to the ground
state $j=0.$
