Since I was running out of space on the comments, I decided to post this as an answer:
Regarding the asymptotic behavior of the spectrum of the Laplacian (or, as you put, the behavior at infinity), the most basic result is Weyl's asymptotic formula (see Chavel's book, p.172): let $N(\lambda)$ be the number of eigenvalues of the Laplacian on a compact manifold $(M,g)$ with $\dim M=n$, counted with multiplicity, that are $\leq\lambda$. Then
$$N(\lambda)\sim vol(M,g)\frac{vol(B^n)}{(2\pi)^n}\lambda^{\frac{n}{2}}, \quad \lambda\to+\infty,$$
where $vol(B^n)=\frac{\pi^{n/2}}{\Gamma(n/2+1)}$ is the volume of the unit ball of $\mathbb R^n$. In particular, if the eigenvalues of the Laplacian are denoted $\lambda_k$ (monotonically increasing in $k$), then
$$(\lambda_k)^{n/2}\sim\frac{(2\pi)^n}{vol(B^n)}\frac{k}{vol(M,g)},\quad k\to+\infty,$$
so the asymptotic behavior of the eigenvalues cannot be prescribed - it has to satisfy the above.
Also, as far as I understand, Colin de Verdière's result is stronger than stated. Namely, given any compact connected manifold M, with $\dim M\geq 3$, and any finite sequence $0\leq a_1\leq\dots\leq a_k$, there exists a Riemannian metric on $M$ such that the first eigenvalues in the spectrum of its Laplacian are $0\leq a_1\leq\dots\leq a_k$.
Finally, I also do not understand what you mean by "spectrum is a quadratic polynomial", maybe you can clarify your question a little better, if the above does not answer it?