I am a junior android developer and I almost finished the alpha version of my first big project. I think that I have good knowledge of java but I am not sure if I organized
No, in your case, both ways will have the same result...
The only thing to mention is, that if you need to receive the response to your request too (may be in the future), you will need to add a Delegate / Callback / Interface to your class, to get the result right back to your calling activity instance... In that case it would be better to create a "non-static instance method" way... But you can add a non-static Method to your Class too so I don't see anything against it.
UPDATE TO COMMENT
Well for example, if you want to provide a ListView with Images... In most cases you first request an JSONArray with your ListView entries, which contains the links to Bitmaps located on the remote Server...
If you download Images Async and put them into the ImageViews in the rows of a ListView (while the user scrolls), it could be possible that images are loaded longer and the ListView will show images in wrong places... For something like that you will need a Singleton Pattern, which will manage the downloads for you... This will not be possible with your class/static Method