To determine the hybridization, we need to examine the geometry and electronic configuration of the central metal ion.
- In \( [Ni(CN)_4]^{2-} \), the nickel ion has a \( +2 \) charge. The \( CN^- \) ligands are strong field ligands and will cause pairing of electrons. The resulting hybridization of the nickel ion in this complex is \( dsp^2 \), corresponding to a square planar geometry.
- On the other hand, \( BF_4^- \) involves a central boron ion with no d-orbitals involved, and it has an sp\(^3\) hybridization.
Thus, the correct complex with dsp\(^2\) hybridization is \( [Ni(CN)_4]^{2-} \), so the correct answer is \( A \).