I am using ffmpeg to convert a 1080p video to MP4 using this command in PHP.
$command = FFMPEG.\' -i \' . $src . \' -sameq -strict -2 -vcodec libx264 -ar 22050 -
The truth is video encoding a CPU and memory intensive job. If you want to do it you need to give the requisite memory for it.
Since you want to convert a 1080p .mov to a .mp4 of same quality I am assuming you jsut want to change the format of the file and not reencode.
ffmpeg -i input.mov -acodec copy -vcodec copy out.mp4
will do it in a jiffy and without any of the memory overheads
If you want your audio codec only changed don't give the acodec part.
If the input video codec does not work for you [so copy is not an option] try doing mpeg4 encoding for vcodec. Cheaper to encode there but I cannot assure you it will fit in your memory requirements always.