The subgroup of the Brauer group generated by (and in fact, consisting of) such division algebras is called "the Schur group". Brauer-Witt theorem asserts that it is given by cyclotomic algebras, so the answer is negative.
I learned this [here][1].

[1]: https://ac.els-cdn.com/S0021869385713646/1-s2.0-S0021869385713646-main.pdf?_tid=61bee0b0-68b9-4270-85a3-6874a46e8b87&acdnat=1527752129_4d7a4ad2e44b41e4e0aa67c40f5ad85a