This depends by what you mean by "national quality institute", if you mean a national standards institute which also acts as a certification body for example British Standard Institute (BSI) the 2 aspects of the body,statutory responsibility and certification body are separated and should not be confused.
Some countries do not have a national quality institute.
Standards in the USA are issued by private bodies.
The Malcolm Baldrige National Quality Award is given to organizations in non profit, health care, business, and education sectors that show excellence. it is sponsored by the National Institute of Standards and Technology.
Chartered Quality Institute was created in 1919.
The motto of Latin American Quality Institute is 'Quality is our mission.'.
National Neuroscience Institute was created in 1989.
National Institute of Health
Latin American Quality Institute was created in 2001.
National Institute of Virology was created in 1978.
National Institute of Engineering was created in 1946.
Armenian National Institute was created in 1997.
National Cancer Institute was created in 1937.
National Institute of Design was created in 1961.
Belgranian National Institute was created in 1944.