No, he was not. Cohen wrote his own account of the history of forcing, The Discovery of Forcing (Rocky Mountain J. Math. 32 (4) (2002), 1071-1100), where he addresses his relationship with Gödel in detail. In particular, he calls him "one of my heroes" (along with Skolem), and mentions some conversations "years later, after my own proof". Nonetheless, it ends with the following:
"Finally a personal remark. I cannot say that I was a friend of Kurt Gödel. We met relatively few times and there was a gulf of age and background that I found difficult to bridge. Yet, my meetings with him were charged with an emotion that was intense, yet difficult to describe. We each traversed journeys that had much in common. I would like to dedicate this talk to his memory."
Cohen was self-taught as a logician. His interest in independence came from discussions on foundations of mathematics with Feferman and Royden at Stanford in 1962. At some point he came to read "Gödel's monograph" The Consistency of the Continuum Hypothesis, actually a compilation of lecture notes for a course given at the Institute for Advanced Study in 1938-39 (Cohen was there in 1959-61). He was even unaware of Shepherdson's follow-up work which showed that inner models could not prove the consistency of the negation of AC or CH, something he rediscovered in 1962.