The only closure on the reservation that I can recall was Gallop, NM.
On the AZ Hwy site it doesn't show any closures.
https://www.az511.com/
From Show Low you'd be taking 77 to I-40. That's not on the reservation. Then if you take I-40 east to 191 north that a major route and I can find nothing on a closure.
If you take 77 north from Holbrook instead of 191 north it is on the reservation so you might want to stop in Holbrook and ask.
Your post heading is for New Mexico. If you're talking about 77 or 191 that's Arizona into Utah.