This is in informal, colloquial, joking language. Yes, people use the phrase "moving party." The idea is that if you can get a group of friends all helping, the work will become fun. "I got roped into..." means "Someone got me to agree to do something difficult." "Roped" suggests a cowboy catching a calf with a lasso, "roping them in." Speaking the word "quote" out loud means "It is called a 'moving party.'" The point is that it is only CALLED a moving party. It isn't really a party. "Party" suggests fun. Helping someone move is work. "...not a thing" is a phrase that's gotten very popular in the US in the last ten years. It means the phrase moving _party_ is false. There is not really any such thing as a "moving party" because moving is not festive or fun. They got me to agree to help them. They called it a "moving party." In reality, there is no such thing as a moving party. Helping someone move is work, not fun.

To be roped into something = to agree, due to strong pressure or persuasion, to do something you didn't want to do, here it's "moving", probably moving house. To be a thing = to be a known and popular concept/practice (informally). Quote, unquote = mimicking writing something enclosed in quotation marks, here to emphasise that the concept is tenuous. The speaker has agreed to be involved in an event, moving house, that they didn't want to be involved in. It has been called a "moving party" in order to persuade them/pressurise them to agree.