It is perhaps helpful to look at the following papers of Fein and Schacher:

>>Fein; Schacher;
Brauer groups of fields algebraic over Q.
J. Algebra 43 (1976), no. 1, 328–337.

>>Fein; Schacher;
Divisible groups that are Brauer groups.
Commun. Algebra 7, 989-994 (1979).

I doesn't have the first paper but according to MathSciNet review of the first item an abelian countable group $G$ is the Brauer group of some field $K$ if and only if $G$ is the direct sum of a divisible group and a torsion group. Moreover $K$ can be chosen to be an algebraic extension of rational numbers (this is mentioned in the second paper).

Also the authors of the following paper <br>
>>Auel; Brussel; Garibaldi; Vishne;
Open problems on central simple algebras.
Transform. Groups 16 (2011), no. 1, 219–264.
<br>

cite a Russian paper of Merkurjev (published in 1985) in which it has been proved that every divisible torsion abelian group is the Brauer group of certain field.