Does the Bugbear’s "Surprise Attack" feature apply on its first round of combat or the players'?

Question

The Bugbear's Surprise Attack feature reads:

Surprise Attack: If the bugbear surprises a creature and hits it with an attack during the first round of combat, the target takes an extra 7 (2d6) damage from the attack.

Is it referring to the bugbear's first round of combat or the players'?

My players were clearing Cragmaw Castle in LMoP. They were in a long combat that went from room to room, then entered another room and got surprised by a bugbear. I wasn't sure if this triggered the surprise attack or not...

I assume it doesn't trigger and it's meant as if the Bugbear ambushes an unaware party, but I'd like to be sure.

Answer 1

It is the same thing

The bugbear and the players each have a turn in the first round.

The Order of Combat (PHB p189):

The game organizes the chaos of combat into a cycle of rounds and turns. A round represents about 6 seconds in the game world. During a round, each participant in a battle takes a turn. ...
Once everyone has taken a turn, the fight continues to the next round if neither side has defeated the other.

In your example, the bugbear had a turn, either waiting for the party to arrive, or approaching them. If it knows about the combat, he is a part of it, for example its movement should be tracked in turn increments.