I don't understand your question: Christian Remling's comment which you referred to points out that if $g$ and $\hat h$ both vanish on a half-line, then $f(x,y) := g(x)\,h(y)$ vanishes on a half-plane and $\hat f$ also does. But vanishing on a half-plane is stronger than vanishing on a quarter-plane, so it seems to me that this $f$ exactly answers the question of your last paragraph.
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