No, it wouldn't
Traunt works regardless. No matter what move is used, traunt will activate on the second turn.
However, I can think of a couple of ways that you could get rid of traunt altogether, but would only work in double or triple battles.
Here they are:
1. You sent out Slaking and Cofagrigas
Slaking used pursuit on Cofagrigas
Then Slakings ability will change to muumy, thus getting rid of traunt!
2. You send out Slaking and Anyone who can use Worry Seed
(pokemon) used Worry Seed on Slaking
And Slakings ability will be changed.
Hope this helped!