Is there any way to export/convert the video (.mov) to .mkv while keeping the same resolution?
Yes. It's called ffmpeg. It's an advanced high level command line tool for basically all your video conversion needs.
Download the tool from here. You'll need to unpack it to any desired location. You can either add the bin folder to path or copy your .mov video file into the bin folder itself.
Then open the command prompt and type in the following code.
ffmpeg -i "filename.mov" "filename.mkv"
The video should now convert to .mkv format preserving the resolution.
.mkv is a container format. Which means the video file can have any codec, including those not supported by TV. So its useful to know which codecs the TV supports. By default, ffmpeg uses libx264, which should work in most devices.
Also check this question, about a similar problem I had with my TV. It should make things a lot more clear to you.
PS: As mentioned in the other answer, Handbrake should also work. Its the graphical tool based on ffmpeg.