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.
National Neuroscience Institute was created in 1989.
Chartered Quality Institute was created in 1919.
The motto of Latin American Quality Institute is 'Quality is our mission.'.
National Institute of Health
National Institute of Sports was created in 1961.
National Institute of Engineering was created in 1946.
National Institute of Justice was created in 1968.
National Institute of Design was created in 1961.
National Institute on Aging was created in 1974.
Institute of National Remembrance was created in 1998.
Belgranian National Institute was created in 1944.