You are definitely using the Quick select tool and not the magic wand? You have the image layer selected in the layers panel?
It should be an easy selection, appart from the narrow stem, but that's doable as well with a small brush size.
I'm not entirely sure what the issue is happening here. I saved your image and opened it into Photoshop. I suspect that you may have the wrong tool selected.
What I did was select the quick selection tool (w). Then make sure you select the 'Add a selection' brush in the top menu.
These are my settings, you can try match it to see if it works for you:
Once selected, I simply clicked on the leaf and it highlighted one side only.
Then, to complete the leaf, I clicked the other side.
You can also UNSELECT parts of the mask by using the 'subtract from selection tool'.
Once you are happy with it, you can copy and paste the selected mask into a new layer and begin working with it further.
I am not sure why when you tried it selected the outer frame. I also suspect it may be because you are in the wrong layer. Since I saved the file as one document, the image is only one layer. However, if you are selecting the black background layer and you click the leaf (which is on another layer) it may be causing this to happen.
Please let me know if this was helpful, feel free to provide more information about your file if you need additional help.