  We obtain exact formulas of the first two cumulants of particle current in
the q-boson zero range process via exact perturbative solution of the
TQ-relation. The result is represented as an infinite sum of double contour
integrals. We perform the asymptotic analysis of the large system size limit
$N\to\infty$ of the expressions obtained. For $|q|\neq1$ the leading terms of
the second cumulant reproduce the $N^{3/2}$ scaling expected for models in the
Kardar-Parisi-Zhang universality class. The scaling
$q\asymp\exp(-\alpha/\sqrt{N})\to1$ corresponds to the the crossover between
the Kardar-Parisi-Zhang and Edwards-Wilkinson universality classes. Under this
scaling the sum converges to an integral, resulting in the crossover scaling
function derived previously for the asymmetric simple exclusion process and
conjectured to be universal.
