The sentence consists of several components, each to be examined for grammatical correctness. Let's break down each part:
I. My cousin brother, who lives
II. in Goa, is eager to visit us
III. in Mumbai and aspires to have
IV. a glimpse of the city
The error lies in Part I. In English, the phrase "cousin brother" is not grammatically correct. The term "cousin" already implies a sibling, so the word "brother" is redundant. The correct usage would simply be "cousin." Therefore, Part I contains the grammatical error. The correct phrase should be "My cousin, who lives."
Hence, the component with a grammatical error is:
Only I