Can two minimum spanning trees for a same graph has different number edges?
The answer is no. When running an algorithm for finding MST, It is first of all a tree, it has |v| - 1 edges $exactly$.

If there are more than |v| - 1 edges, than there're two vertices that their path isn't simple, which is simply shown with induction. Meaning if there's a path from s to t with more than |v| - 1 edges, one of the vertices will be repeated.