What is the minimum number of overs required to bowl in the second innings for applying Duckworth-Lewis Method to find the winning side?
After the team batting first has started its innings and there is an interruption due to rain or any other factor, Duckworth-Lewis Method can be applied. This DL algorithm is applied and the target for the team batting second is calculated based on the overs that can be bowled as decided by the umpires.
If I understand your question correctly, you are asking when the result of the game will be decided by DL method and not when DL method is applied. This is done only when the side batting second cannot continue its innings due to the interruption and has completed atleast 20 overs (ODIs) or 5 overs (T20s).
Duckworth-Lewis can be applied at any stage: the tables allow for the calculation of the par score after each ball, so a team could theoretically win on D/L after just one ball of their innings.
However, in order to prevent teams winning on the basis of such a small "sample size", to borrow a statistical term (for instance, winning on the strength of hitting the first ball of the innings for 6), playing regulations (such as in international cricket) specify that a minimum number of overs must have been completed in each innings for a result to be decided. As @Orangecrush states, in 50-over ODIs this is set at a 20-over minimum, and in 20-over T20Is at 5 overs. As the ICC Playing Conditions state, in section 12.4.2:
and for T20I: