The answer is: of course not. Indeed, let $a=1$, $b=0$, and take any real $m>0$. If $f$ were analytic in $x,y$ such that $-m\le\Re x,\Re y,\Im x,\Im y\le m$ then it would be analytic in $x$ at $x=0$ for $y=0$. Then the function 
$$\mathbb R^2\ni(s,t)\mapsto\frac{Q(s)}2\,\ln\frac{Q(s)}2$$
would satisfy the [Cauchy–Riemann equations][1] in a neighborhood of $(s,t)=(0,0)$ -- which it does not. 


  [1]: https://en.wikipedia.org/wiki/Holomorphic_function#Definition