No. Let w.l.o.g $Q:= (0,1)$. There is a bounded linear functional $f$ on $E$ such that for any $x\in E$ one has: $\liminf _ {s\to 0} x(s)\le f(x) \le \limsup _ {s\to 0} x(s) $. This functional is positive, still vanishes on some functions which are strictly positive on $Q$.
rmk. For the construction of $f$, you may directly refer to the Banach limit functional $\phi:\ell^\infty\to\mathbb{R}$ and define $f$ by composing it with the linear bounded map $E\to\ell^\infty$ taking $x$ to the sequence $\{ x(1/n): n\in\mathbb{N}_ + \}$. You can adapt this construction to more general non-compact $Q$.
