As noticed by Sasha in his comment, the answer is no. Let me give some more details on the proof.
Given a smooth complex hypersurface of degree d, say $X_d \subset \mathbb{P}^{n+1}$, by standard arguments involving Lefschetz theorem we have
$H^k(X_d)= H^k(\mathbb{P}^n)$ for $k \neq n$.
In particular, all the odd Betti numbers are zero, except possibly the middle Betti number when $n$ is odd. On the other hand, the Euler-Poincare characteristic of $X_d$ can be computed and it is equal to
$\chi(X_d)=\frac{1}{2}[(1-d)^{n+2}-1]+n+2$,
so for $d=2$ and $n$ odd the middle cohomology group must be zero too.