Matrices range relation with a symmetric positive definite matrix

You may take $Q=WW^T+I-V(V^TV)^{-1}V^T$. First, since $QV=WW^TV$ and $W^TV$ is invertible, $QV$ has the same range as $W$. Second, as $Q$ is the sum of the Gram matrix $WW^T$ and the orthogonal projection $I-V(V^TV)^{-1}V^T$, it is positive semidefinite.

Finally, if $x^TQx=0$, then $x^TWW^Tx=0$ and $x^T(I-V(V^TV)^{-1}V^T)x=0$. The latter equality implies that $x$ lies inside the range of $V$, so that $x=Vy$ for some vector $y$. The former equality thus implies that $y^TV^TWW^TVy=0$. As $W^TV$ is invertible, we must have $y=0$. Therefore $x=0$. Hence $Q$ is actually positive definite.