It might take years to get a quest with a ress serum, since it's random. The best way is to start exploring every single tile near your base, by settling, searching the ancient dangers, and repeating until you find one.
The absolute fastest way is to send several parties to different tiles at once. Build a steel door next to the wall of the danger, then deconstruct it and run. You'll see inside and if there is no mech serum just leave the map and abandon the tile. If there is one, leave the map but don't abandon the tile. You can come back in force and deal with the mechs or bugs. This of course sacrifices the other valuables you might find, but it's faster.
The absolute fastest way is to send several parties to different tiles at once. Build a steel door next to the wall of the danger, then deconstruct it and run. You'll see inside and if there is no mech serum just leave the map and abandon the tile. If there is one, leave the map but don't abandon the tile. You can come back in force and deal with the mechs or bugs. This of course sacrifices the other valuables you might find, but it's faster.