"Grain" typically means the seeds of cereal grasses, of the family Poaceae. Quinoa is the seed of a plant from family Amaranthaceae; it's not a grass, it's a leafy plant from the same taxonomic group as beets and spinach. This leads to it being referred to as a "pseudocereal grain" if it's called a "grain" at all.
cereal grains (from the Poaceae) actually are both a seed and a fruit. In most grasses the ovary wall (fruit) and the seed are fused together into a structure called a caryopsis. Sometimes the ovary wall is milled off before consumption (often with wheat), sometimes not (corn, oats). Grain is not a well defined term, at least botanically. Soybeans, corn and wheat are all considered grains by some (for example, some departments of the US government).
