Need an Identity for CTFT Polynomial Raised to an Exponent
Dilip's idea is an excellent suggestion. Let me slightly modify it. You can rewrite
$$a\omega ^{2}-b\omega +c = a(\omega -\omega_{0} )^{2}+d$$
So the whole thing can be written as 

$$e^{d}\cdot e^{a(\omega -\omega_{0} )^{2}}$$

That's essentially a Gaussian shifted in frequency with a constant scale factor. The transform of a Gaussian is just another Gaussian and the shift in one domain corresponds to multiplication with a complex exponential in the other domain so the inverse ought to look something like:

$$x(t) = k\cdot e^{\tilde{a}t^{2}}\cdot e^{j\omega _{0}t}$$
where $k$, $\tilde{a}$, and $\omega _{0}$ are determined by the original a, b, and c. a better be a negative number, though, otherwise I don't think that the transform would be convergent.