As QGIS is a GIS it knows how to handle data of different coordinate reference systems. It can overlay them or transform them as needed. All as long as their metadata is actually correct of course.
To turn your map visualisation into a different projection you can choose a different CRS in the bottom right corner (it should say "EPSG:4326" for you ...
I saw this post while I was looking for the U.S. National Grid System. I think this might work for you. I agree with the person above about the coordinate system, BUT if you want grids across the United States this could be a starting place. https://usngcenter.org/portfolio-item/usng-gis-data/
GIS - Survey123 Jedi
A >>>Custom Affine Projection<<< would work well in QGIS. You'd need 3 Match-Up Points and a little work in Excel to get A0,A1,A2,B0,B1,and B2 numbers.
Attached here is a typical spreadsheet:
It's good idea I have done that before.
You just need 2 points in UTM CRS to move your all of your work
if you have more 2 points calculate the conformal transfer parameter by least squares than use the calculated shift and rotation to move your work.
.img files are in a Garmin proprietary format. The format has been reverse-engineered and there are tools that will allow you to generate your own .img files, though it is not for the faint of heart. The best explanation I've found is here. This covers how to generate a topo map; the grid you want to produce should be simpler, though the key step of ...