In Wikipedia it states that a quasi-coherent sheaf on a ringed space $(X, \mathcal O_X)$ is a sheaf $\mathcal F$ of $\mathcal O_X$-sheaf of modules which has a local presentation, that is, every point in $X$ has an open neighborhood $U$ in which there is an exact sequence: $$ \mathcal{O}_X^{\oplus I}|_{U} \to \mathcal{O}_X^{\oplus J}|_{U} \to \mathcal{F}|_{U} \to 0 $$ for some (possibly infinite) sets $I$ and $J$.
Is the $I$ and $J$ have to be fixed for all open $U$ or can they depend on $U$? Clarification is appreciated. Thank you.
They can (and will usually) definitely depend on $U$.
Take the simplest example: $(X,\mathcal{O}_X)$ is the spectrum of $R=K_1\times K_2$ where the $K_i$ are fields. The topological space $X$ is then discrete with two points: $x_i$ with residue field $K_i$, for $i=1,2$. Write $U_i=\{x_i\}$.
Now giving a $\mathcal{O}_X$-module $\mathcal{F}$ is the same as giving a vector space over each $K_i$: $\mathcal{F}_{x_i}=\mathcal{F}(U_i)$ is a vector space over $\mathcal{O}_X(U_i)=K_i$. Let us write $V_i$ for that vector space.
In particular, you get $$0\to (\mathcal{O}_X^{\oplus J_i})_{|U_i}\to \mathcal{F}_{|U_i}\to 0$$ where the cardinal of $J_i$ is the dimension of $V_i$. (So $\mathcal{F}$ has to be quasi-coherent.)
If $V_1$ and $V_2$ have a different dimension, you need $J_i$ of different sizes.