A: Whose treat is today?
B: It is my treat. Let me treat you to lunch.

Is the above dialog grammatical?
Whose treat is today? is not correct.

A. Whose turn is it to treat today?
B. Mine. / It's my turn to treat you to lunch.

More usual:

Let's go to lunch. It's my treat. (There's no sense of taking turns in this version.)

Teachers: We supply a list of EFL job vacancies
A: Did you go Dutch yesterday evening?

B: No.

A: Whose treat?

B: His treat.

Is the above dialog acceptable?
Yes. I'd say so.