In this article, using 3-orbifolds singular along a knot with underlying space a homology sphere $Y^3$, the question of existence of non-trivial and non-abelian $\SU(2)$-representations of the fundamental group of cyclic branched covers of $Y^3$ along a knot is studied. We first use Floer Homology for knots to derive an existence result of non-abelian $\SU(2)$-representations of the fundamental group of knot complements, for knots with a non-vanishing equivariant signature. This provides information on the existence of non-trivial and non-abelian $\SU(2)$-representations of the fundamental group of cyclic branched covers. We illustrate the method with some examples of knots in $S^3$.