No, they are completely different. "General" is a commonly used word; "generic" is rare.

"General" has a lot of different meanings, but as an adjective it means "wide," "covering everything," "including everything."

"The old-time general store sold everything--food, clothing, and hardware."

"In general, the United States has a temperate climate, but there are many exceptions."

"Massachusetts General Hospital treats everything; the Dana-Farber Cancer Institute specializes in oncology."

"Generic" means "the plainest and most ordinary of its kind." Often, it means "lacking a brand name." 

"Don't pay extra for brand-name aspirin; aspirin is aspirin; just buy the cheapest generic aspirin."