In Fiji, you can save images in OME-TIFF formats using Bio-Formats Exporter either from File > Save As > OME-TIFF… or Plugins > Bio-Formats > Bio-Formats Exporter.
In the file save dialog, you can choose many file formats (jpeg, jpeg-2000, OME-TIFF, OME-XML, QuickTime, TIFF, Vaa3d, Woolz).
If you need OME-TIFF, choices are .ome.tif, ome.tiff, ome.tf2, ome.tf8, ome.btf (I noticed that in macOS the list is truncated and you can’t really see all of them).
I looked for the differences between these file extensions in the ImageJ web site or Bio-Formats documentation, but I could not find any clear descriptions about them. In the BigTIFF web page, they mention .btf and .tf8.
I figured out that by specifying .ome.btf extension instead of the default .ome.tif, you can save an image into 64-bit BigTIFF format rather than usual 32-bit TIFF file, the latter of which has an upper limit of 4GB in file size. This is the simplest way to use BigTIFF format for OME-TIFF without using Bio-Formats Java API and no page seems to mention this method.
I wonder it might be worth adding this information to the following page under the section " Bio-Formats Exporter".
Still I have no knowledge about .ome.tf8 and .ome.tf2, though I guess .ome.tf8 might be equivalent of .ome.btf. Can someone tell the difference?