Is desire a Common Noun

Yes, the noun desire is a common noun; a Word for any desire of any kind.

A proper noun is the name of a person, a place, a thing, or a title; for example:

• Andre P. Desire MD, Cardiology Wichita Falls, TX
• Desire Street, New Orleans, LA and Desire Avenue, La Puente, CA
• Hearts Desire Inn, Raton, NM
• "A Streetcar Named Desire" by Tennessee Williams