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
Auel; Brussel; Garibaldi; Vishne; Open problems on central simple algebras. Transform. Groups 16 (2011), no. 1, 219–264.
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.