If I have Darius Trifarian on field and I play Heedless Resurrection as my first card of the turn, would the card played by Heedless Resurrection count towards Darius' ability?

Yes — the unit played by Heedless Resurrection counts as the second card played, so Darius triggers.

Sequence: Heedless Resurrection resolves as card 1, then plays a unit from your trash during its resolution. That unit goes on the chain and resolves — once it does, the act of playing it is complete, and Darius (still on the board) sees it as your second card played this turn and triggers (+2 Might, ready). Cards played by game effects are full "plays" (Rule 419.3, 419.4.a), and this is consistent with rulings that cards played via abilities still count as played cards.

One caveat: if the resurrected unit is countered and never resolves, it doesn't count, and Darius won't trigger (Rule 419.4.a.1).