Can wp_get_image_editor convert to icon?

No. At least I’ve never saw it.

Although I believe that you can tailor some solution. Since WordPress 3.5 in WP_Image_Editor is added support for Imagick by default (which is a wrapper to the ImageMagick library). You can check class-wp-image-editor-imagick.php to see how it’s integrated and if you have ImageMagick installed on your server, probably you can tailor something. But I believe it will probably be an overkill for a small set of .ico files.

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