Does Adapter have a Direct Stream Copy option?
That way you don't have to actually convert the video & audio streams ( = quality loss). There is no reason to convert the actual streams.
The video has most likely already been MPEG4/H264 compressed when the MKV file was created.
A really good conversion tool will allow you to simply change the file container without transcoding the audio & video streams again - plus it saves quite a lot of time if it's a long video.
TEncoder has this feature - and it's also free.
But of course: You still have to bear in mind that the MP4 container is much more restrictive than the Matroska container, so you might be forced to convert anyway.
Lack of MKV support is usually an indication of commercial arrogance.