Does the flu start as a head cold no fever or cough?

Sounds like a Cold: I typically find that the actual "flu" - the disease caused by the Influenza Virus - starts as a rapid onset of fever and chills, often with a dry, non-productive cough, and usually accompanied by chills, body aches, and fatigue. What you are describing sounds more like a viral cold, or perhaps even an allergy to something, considering the lack of fever.