Why are the letters 'Q' and 'Z' missing from the telephone buttons?

They aren't on mine. Must be one of those things associated with certain models and companies...

My guess would be that in the days when exchanges had names, and the names were spelt out using the numbers letter function, Q and Z would have been in so few names it wasn't worth featuring them as redundant letters on the dial (this was long before we had buttons).

In London each exchange had a name made from the available letters till they all ran out, when they switched to numbers to use all the combinations that wouldn't have made a word.

Satguru's answer is what I found also. The Q was even more limited in use because most Q words are following by U limiting the exchange even further. We are really aging ourselves when we can remember numbers like VIking 6-0000 or ATlantic 4-0000.

https://www.straightdope.com/21341529/why-is-there-no-q-on-the-telephone-dial

PEnnsylvania 6-5000!

We did have a discussion of telephone number issues not too long ago. Gmack was interested in 555 numbers.