The Search for Barta
To prevent having to come back, go ahead and buy a Hearty Durian from the fruit vendor if you do not already have one.
Return to the objective marker and talk to Liana to complete the quest and get a Silver Rupee.
Barta has disappeared again. She headed south in search of leviathan bones but never came back. Not only is the area home to Molduga, but it's also known for extreme heat. You'll need special gear, or you'll be in danger. Use your map to search for Barta.
You found Barta collapsed in the desert.
She keeps muttering, "A hearty durian, a hearty durian," over and over again, deliriously.
...Seems like she really wants to eat a hearty durian.
Updated Quest 2
You fed Barta a hearty durian after finding her collapsed in the desert.
She seemed much better, but once you told her of Liana's anger, she dashed off back to Gerudo Town. Just in case, you should let Liana know that Barta is OK.
Barta seems to have made it back to Gerudo Town in one piece.
But it sounds like Liana really chewed Barta out for making her worry. Maybe she'll lay low for a little while.