Factoring out whichever of $a,b,c$ has the smallest valuation, you can assume (I think) that at least one of $a,b,c$ is a unit. If $a$ is a unit, then
$$
  ax^2+bxy+cy^2 = a\left(x+\frac{b}{2a}y\right)^2 
   - \frac{b^2-4ac}{4a}.
$$
Similarly if $c$ is a unit. If $a$ and $c$ are non-units and $b$ is a unit, then the "$a$" coefficient of $f(x+y,y)$ is a unit. This seems to give your claim with the stronger result that $a'$, $b'$ and $c'$ all have valuation at least $\nu$. (Maybe I'm missing some subtlety here about general valuation rings?)