When is Teacher Zhang's birthday?

The answer is June 4th.

Xiao Ming said: If I don't know, Xiao Qiang certainly doesn't know.

This sentence means that you can lock the birthday time in

June 4, June 7

September1September 5

Xiao Qiang said: I didn't know at first, but now I know.

June 4th and September 1 will be fine (in Xiao Ming's view).

Xiao Ming said: Oh, I know that, too.

Locked on June 4.