I'm getting rid of the quick release and hollow axle so that I can use a solid axle with some track nuts. How do I remove the hollow axle and replace it with the solid axle?

what hub?
generic cup and cone? if so just undo the locknuts, unscrew the cones and it'll slide right out.
if its a cartridge bearing setup it might require whacking it with a hammer

It's a fairly new Shimano Deore hub. I think Shimano uses a cup & cone bearing system for all of its hubs, so I think mine is a cup & cone type of hub. Is there a page on the internet somewhere with some pics of how to do this? I tried searching for axle removal/installation on the internet, but all I got was a bunch of maintenence info for car axles.

Yup, shimano axles are 9x1mm, if I recall correctly. Just about evry bike shop should have a replacement axle. And yep, undo the locknut, the conenut, then the axle and bearings will pop out, and replace the axle.

Is this a front hub? It's slightly more complicated with a rear hub, but both are pretty straighforward.

If you are at all familair with cone adjustment, this will all be very simple. Here's the instructions for a bearing rebuild:
https://parktool.com/repair/readhowto.asp?id=105

here:
https://"https://www.parktool.com/repa...to.asp?id=105"

park tool's website has a lot of useful instructions under the repair help section

edit:beat me to that ****.

Hey thanks guys. This is just what I needed. I got the solid axle and wheel in the garage. Now, off to the bike shop to buy some cone wrenches.