Let's look at the breakdown:
* "In" means "into" or "within."
* "Trud" comes from "trūdere" (to thrust, push, or shove).
So, "intrude" combines the concepts of "entering" and "pushing" or "forcing" into a situation.
While not directly related to "push," the word "intrude" does stem from a root that encompasses the concept of pushing or forcing.