When you use Dialga GX's "Timeless GX" attack does Poison still happen in between turns?
Nope, the attack says to "skip the between-turns step", so Asleep, Poisoned, Burned, and any other between turn effects also get skipped."
Source: Ultra Prism FAQ (2018-01-19), TPCi Rules Team (2018-02-01)

Relevant cards

Note: All applicable cards may not be listed below.
SM5_100
SM5_146
SM5_164