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