A double-stranded DNA (dsDNA) is modeled by two coupled one-dimensional Kitaev's chain and the topological superconductivity is studied. It is shown that the zero energy mode exists under some specific conditions. The wave function of zero mode is calculated and it is shown that the Majorana quasi-particles exist on the ends of each strand. By calculating the winding number, we show that the topological phase transition can happen if the hopping integral between two strands is very smaller than the pairing potential between the Cooper pairs. It means that the dsDNA behaves as a trivial superconductor, commonly, but single-stranded DNA (or two coupled ssDNA with very small hopping between them) may behave as a non-trivial superconductor. Finally, we suggest an experimental setup for probable detection of Majorana quasi-particle in DNA.