Consider the definition of a graph as $(V, E)$.
The first element is a set of vertices and the second one is an set of edges (relation over $V$).
My doubt is how the order of elements impact the definition. If I consider as $\{V, E\}$, then what will be the problem?
If the answer is we should have a set of vertices to make a relation(edges) over them. Then consider some other definitions such as FA in which $q_0, F$ can come just after $Q$ instead of $\delta$ right? Are they important in programming aspect?
It seems that you already know the answer - the two sets are not of the same type; One is a set of vertices, the other is a relation. You can't have the set of vertices act as a set of edges, since it doesn't define a relation on the other set.
The fact that the set of vertices comes "before" the set of edges in the ordered pair $(V,E)$ is a convention - it doesn't matter who comes first, as long as everybody knows what you're talking about. For the sake of convention, the set of vertices is written first, in order to clarify what set the relation is defined on, but it really doesn't matter. The thing that matters is that there is one set that consists of vertices, and another set that defines a relation - and naturally, they cannot "switch parts".