The story
From quad (“four (“4”)”) + bike (“bicycle”). From being a four-wheeled vehicle (a quadricycle) that is ridden astride like a motorcycle, the term carries an inherent contradiction, since “quad” denotes four, while “bike” implies two. This etymological inconsistency is largely overlooked in everyday usage, as such contradictions are often absorbed and normalized by idiom.
Etymology from Wiktionary, CC BY-SA 4.0.