Had to take your icon from the image you posted, so a bit on the small side.
1. Make sure the image is in RGB mode Image -> Mode RGB
2. Make sure the image has transparency: Layer -> Transparency -> Add Alpha Channel (if greyed out it already has an alpha-channel)
3. Use the fuzzy select tool,
click in the white background to create a selection.
4. Grow the selection by one or two pixels Select -> Grow
should look like this: https://i.imgur.com/PcDad4P.jpg
5. Use the Color-to-Alpha filter Colours -> Colour to Alpha using default white.
looks like this: https://i.imgur.com/m5gz3W0.jpg
6. Ok that, kill the selection Select -> None,
Export as a png file which will keep the transparency.
1. Make sure the image is in RGB mode Image -> Mode RGB
2. Make sure the image has transparency: Layer -> Transparency -> Add Alpha Channel (if greyed out it already has an alpha-channel)
3. Use the fuzzy select tool,
click in the white background to create a selection.4. Grow the selection by one or two pixels Select -> Grow
should look like this: https://i.imgur.com/PcDad4P.jpg
5. Use the Color-to-Alpha filter Colours -> Colour to Alpha using default white.
looks like this: https://i.imgur.com/m5gz3W0.jpg
6. Ok that, kill the selection Select -> None,
Export as a png file which will keep the transparency.