Does the Catcher in the Rye belongs to the beat generation?
That is a good question. I think the best way to answer your question is to look at some dates. The beat generation refers to an American post Word War II movement in the 1950s. This movement espoused freedom in various areas, such as experimentation with drugs, rejection of things mainstream, like materialism, and a foray into Eastern religion. Allen Ginsberg and Jack Kerouac are some of the most famous writers of this movement.
The Catcher and the Rye was published in 1951 and became a instant success. The main character Holden leaves his private school in Pennsylvania in view of all the superficiality and phoniness of people. He heads to New York and spends time with various people like a prostitute and strangers. He is the embodiment of angst, alienation, confusion, and a desire to know what to do with his life. At one point he wants to leave everything behind and go out west, which he does not ultimately do.
From the point of view of chronology, the Catcher and the Rye is contemporaneous with the beat movement. Some of their influences were in common. We can be sure of this, because they emerge from the same cultural context. However, to say that the Catcher and the Rye influenced the beat movement (or vice versa) is not something that we can say. Holden of theCatcher and the Rye is a very tame version of the proponents of the beat movement.
So, in short, we can say that there are some common themes and it belonged to the same intellectual milieu, but there were not the same movement.