"fat tyres"

I hope I am in the correct forum - "fat tyres"!
Will a 26x1.75 tyre fit a 26x2.10 rim?

Now before you laugh, I do not own a mountain bike (this is for a friend) and I have always had the same size road wheels hence the choice of user name.

There are actually several different 26" size rims! You have to check the ERTO on the existing tyre & the new tyre.

Otherwise, you should be "okay" putting the narrower tyre on the current rim which is hopefully a 559x17, a 559x22, or something in between.

BTW. You'll probably need to use different tubes for the two different tyre sizes.