does deathkknell trigger first before the battlefield trigger in riftbound if the unit dies in that battlefield?

No — there's no built-in "Deathknell first" rule. If the battlefield's ability triggers off the same death event (e.g. Black Rose Sanctum's "first time each turn a non-token unit they control dies here"), both triggers are simultaneous, and whoever controls them picks the order they go on the chain (Rule 383.3.d). Chain resolution is last-in-first-out, so the trigger placed last resolves first.

That means the practical answer depends on control. Usually you own the Deathknell unit and control the battlefield, so you control both triggers and can choose — decide which one you want to resolve first and place it on the chain last. If different players control them, the turn player orders their trigger first, so the non-turn player's trigger is placed last and resolves first (Rule 383.3.d.1). Deathknell being recorded as a Pending Item before the unit hits the trash (Rule 808.1.d.2) does not give it priority on the chain.

One distinction worth flagging: if by "battlefield trigger" you mean a conquer/hold ability like "when you conquer here" or "when you hold here," that's not simultaneous at all — control is only established after the showdown/combat cleanup, and Deathknell resolves out of that cleanup, so Deathknell always happens before the conquer/hold trigger (FAQ #10444).

If you name the specific battlefield, I can tell you who controls its ability and therefore whether you actually get the choice.