No, Rosie Cotton does not appear in The Fellowship of the Ring.
While she is a beloved character in The Lord of the Rings legendarium, her role is primarily more prominent in The Two Towers and The Return of the King. Rosie is known as the love interest of Samwise Gamgee, and her character adds depth to Sam’s motivations during the quest. In The Fellowship of the Ring, however, she is mentioned only briefly in relation to Sam’s thoughts and feelings, but she does not physically appear in the story.