This is done from the "File" menu in any drawing window or in the parameters window. If you are using the parameters window menu the picture that will be saved is that of the current drawing window, that's the one with the star infront of its title.

To save a picture you must first decide if you want to save it to a file on your computer or to a URL. This is controlled with the "File" menu's "IO to URL?" item. If it is checked the reading and writing of fractals is done with URLs.

With that item set as desired, select "Save Image..." to save a picture. Pick "Open JQ Image..." to read in a fractal that was generated by JavaQuat.

JavaQuat fractals are saved in a file that not only contains the picture, it also contains all the arguments from the parameters window and menus that went into creating the picture. JavaQuat expects to see these when it reads in a fractal picture and will complain if they are not there. Having all the arguments means you can continue wirking with the fractal, zooming in on details or whatever, as if you had just drawn it.