Just enter the training PvP queue. It's just that. According to what I remember of @Senshu saying on Global, the new client will have some hooks in allowing that quest to count skirmish battles as well.
Thanks for asking about this. This is unfortunately one of the transitional issues at the moment as we start getting things on the live server ready for the new client. We needed to revise the Training Day daily because it didn't work in the new game client. Right now if you create a custom match and win you should get credit but only PvP. The PvE won't work until the new client is released.
