This is mentioned in Ueno's Algebraic Geometry 3, Chpt 7, Sec 2 without concrete example. It is located at pg 43 right above exactsequence (7.33).
Let $X$ be a scheme and $U=\operatorname{Spec} R\subset X$ open affine. Define $Q(U)=S^{-1}R$ where $S$ is the set of non-zero divisor of $R$. This defines a pre-sheaf over affine open set of $X$. Then I can sheafify it to get $Q$ sheaf of total quotient over $X$. Computing stalks would not be too hard from definition. However, I have significant amount of hard time to even see what is $Q(Spec(R))$ or $Q(U)$ over $U$ affine.
$\textbf{Q:}$ What are non-trivial computable non-integral examples of such $Q$? I am looking for some computable examples to see what kind of sheaf I am dealing with. It is easy to deal with single point or finite number of points. I am looking for examples of at least dimension 1 above.
My comment was not adequate, so here is a clarification. The assignment $U \mapsto Q(U)$ that you describe is not a presheaf for $X$ nonintegral (when we allow ourselves to consider nonaffine $U$). This has something to do with the fact that an arbitrary map of commutative rings does not send nonzero divisors to nonzero divisors in general. Consequently, this object will be fairly complicated. For more information about this, see the paper $\textit{Misconceptions about $K_X$}$ by Kleiman.
When $X$ is integral, the situation is nicer. In this case, it is trivially true that nonzero sections of the structure sheaf restrict to nonzero divisors, so the sheaf that you describe is well-defined, even over nonaffine $U$. When $X$ is affine, the global sections of $Q$ will just be the fraction field of $\mathcal{O}(X)$.
Remark: When $X$ is noetherian, the value of the sheafification of $Q$ at an affine open $U$ being the total ring of fractions of $\mathcal{O}(U)$ has to do with the fact that we are sheafifying. If $X$ is not noetherian, the value of the sheaf on an affine open may be different from that of the presheaf $Q$. For what it's worth, in my limited experience I've never seen anyone use this sheaf for schemes that are not varieties.