If a player plays a card that is Defied, will that count as a card played for activating Darius Trifarian?

No, a card that is defied (countered) does not count as a card played for the purposes of Darius, Trifarian's ability.

According to Rule 425.1.b:

"A card that is Countered is not considered to have been played."

Additionally, Rule 419.4.b clarifies:

"If a game effect prevents the resolution of the card—for example, because the card was countered—the card wasn't played and no abilities that trigger on playing cards trigger."

Sequence of events:

1. You play a card.
2. Your opponent plays Defy in response.
3. Defy resolves and counters your card.
4. Because the card was countered, it is negated and cleared from the chain; it is never considered "played."
5. Since the condition "When you play a second card" requires a card to successfully resolve and enter the board, Darius, Trifarian will not trigger if a previous card was countered, as that countered card did not increment your "cards played" count.