I wonder if the strong force can be explained by quaternions without the use of matrices.
I heard that it could, but the source was not reliable.
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Your source probably misunderstood.
Furthermore, as CuriousOne comments, elements of an n-dimensional field can be represented as nx1 matrices (n=4 for quaternion). The weak force can be explained with quaternions instead of matrices, but underneath it's really just the same math in different clothing.
Describing the strong force requires at least 8 independent terms, but quaternions only have 4. Maybe your source meant octonions.