I guess you could use GM:S editor to remove the background around the coin, but it'll likely be a menial and painful task. You'd need to enter the image editor and use eraser tool for that.
Other graphics editors might be better at removing the background around the object, but I can't know of any editor I could recommend for that.
I wonder about the image with transparent background you mentioned - if it's a simple PNG image, then it should be imported to GM:S just fine without any white background. If there's a problem, I'd sooner suspect an image editor like Paint, which doesn't handle transparency and removes transparency from saved PNGs.
Generally, if you import images made with external editor into GameMaker, your best bet is to have them prepared beforehand so that they're good to use in the game; this includes having correct transparency, too.
If you have the image editor source file (rather than exported bitmap), then maybe there's a way to export the image without the background from editor itself? E.g. in Inkscape I could manipulate the visibility of background and foreground layers and export only the layers I need.
If this bitmap is all you have, then you'll need to get your hands dirty and remove the background yourself - whether in GM:S built-in editor or in some other tool.
Once you do have an image with transparency, you can use Image >> Auto Trim All Frames from GM:S sprite editor to remove the transparent framing around the image (it's a little more complex with multi-image sprite).
If you collaborate with an artist, ask them to export the image without the background, just the transparency around the coin. I'm pretty sure any video game artist worth their salt would setup their image in a way that the background can be easily removed, leaving only the transparent area around the main object.