I am a Software Engineer. How do I perform market research on whether there is demand for my product?

You don’t sound like a software engineer. Software engineers care about building software, and don’t give a fig about “market research”. Do you use your software yourself? Why not? Maybe you should build something that you need yourself, because then there would be no question about whether there is “demand for your product”.

I prefer questions from people who build software because it’s fun and useful, not because they’re looking to make a buck.

Huh.. May be I should not have expected much from this site.
I don’t think there is anything wrong with making a buck.
May be you are someone who can’t/won’t/don’t like or don’t know how to make a buck with your creations
May be you are someone who has infinite time or prefer to just build software nobody needs – just because it is useful to you or whatever.

Try getting some people in the industry to try it out. Look for someone who has been around, and get them talking about what problems they have. If you find a person who is well connected in the industry they will probably give you a good idea of whether your product has any value or not. If they like what they see they will also likely be able to get you in front of others.

Also, if the problem your product solves is measurable and industry figures are available you can get a potential market. You can also try to have people fill out surveys, but that can be pretty difficult and sometimes worthless if not done correctly.