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A torus-covering $T^2$-knot is a surface-knot of genus one determined from apair of commutative braids. For a torus-covering $T^2$-knot $F$, we determinethe number of irreducible metabelian $SU(2)$-representations of the knot groupof $F$ in terms of the knot determinant of $F$. It is similar to the result dueto Lin for the knot group of a classical knot. Further, we investigate thenumber of irreducible metabelian $SU(2)$-representations using Fox's$p$-colorability.

Torus-covering knot groups and their irreducible metabelian $SU(2)$-representations