Once you have freed the queen you can find her back in Flamerock Refuge. For this Trophy you will need a total of 15 Eridium.
When you approach and talk to her you will get a mission to pet her. After you completed that you will get a mission to feed her. This will cost you 5 Eridium. After the completion of that mission just stay there and feed her two more times with 5 Eridium each.