Welcome to the forum!
The proper way to export a PNG with transparency, is to use File menu > Export PNG. If that's already what you're using, you might not be using the right options there. First notice the 4 choices at the top of the dialog (Page, Drawing, etc.). If you want the whole grid, regardless of where the page borders are, choose Drawing, and that should export the entire grid.
But if you have other drawings on the page, you might need to select only the grid, and use the Selection option.
Also, be sure to click Export As, which allows you to both choose which directory you want to export to, as well as choose a name for the file. If you don't choose that, Inkscape will give it a weird name, and put it wherever it wants (usually your user directory, which I don't know about you, but that's definitely not where I want my images).
I'm not sure how to make it high res, for import into your other program. If you change the DPI setting in the Export PNG dialog, it will make the image larger (if you make the DPI larger). For printing, when you set the size for the print, then all that higher dpi gets compressed back down to the print size, and you end up with a higher res print.
But for your othe program, I'm not sure how that would work. Hopefully some other members will be along soon, who can explain that for you.
