The invention relates to the novel cotton variety designated 03Q066. Provided by the invention are the seeds, plants, plant parts and derivatives of the cotton variety 03Q066. Also provided by the invention are tissue cultures of the cotton variety 03Q066 and the plants regenerated therefrom. Still further provided by the invention are methods for producing cotton plants by crossing the cotton variety 03Q066 with itself or another cotton variety and plants produced by such methods.