Basic Rules, page 72. There's no such thing as an "active perception check" though.
So under Actions in Combat
Search
When you take the Search action, you devote your
attention to finding something. Depending on the
nature of your search, the DM might have you make
a Wisdom (Perception) check or an Intelligence
(Investigation) check.
So to the OP, assuming the enemies were shifty goblins with a bonus action to hide the way I would have run it:
DM: announces "You hear some noise in the bushes, roll for initiative." You can be in combat even if you don't know where the enemy is and no one has attacked. All it takes is awareness of a threat or the intent to attack. This could have also been run as if the PCs were surprised and not able to take actions (or reactions) until their turn.
PC1: Moves towards the bushes and searches for the enemy
DM: Asks for a perception check.
PC1: gives result - this is a contest. PC 1 Perception vs the goblin's Stealth. The goblins win the contest, and are still hidden.
Goblins: attack PC 1 with advantage. This reveals their location at the moment they fire but they can move and take a bonus action to hide.
If they
don't have a bonus action to hide they can still move and the PCs may or may not know exactly where they are. This is kind of up to you and whether or not you think the PCs would know where they are even if they can't be seen. Think of the hide (bonus) action as an action penalty, they're trying to move carefully so as to not make a lot of noise/move bushes in a way that would reveal their location/accidentally step into a gap in the bushes and be seen.
For example, if there's a series of tunnels on the side of the road the goblins could have ducked into the tunnels after firing. The PCs may have even seen them go into the tunnels but would not know where in the tunnels they are.
PC2: Attacks the location they saw the bolts come from. There's nothing there so nothing is hit
Initiative continues as normal.