Let's look at the ratio of two adjacent $q$-binomials as we move away from the center, for simplicity I'll do the even case.
$\binom{2n}{n-a}_q / \binom{2n}{n-a-1}_q = \frac{[n+a+1]_q}{[n-a]_q} > q^{2a+1}$
In particular these are decaying faster than geometrically. With some possible exceptions for say $q =2$ or $3$ and $n$ small, it looks like the middle binomial coefficient will be bigger than the sum of all the others.