I know in Islam there is the belief that Allah does not have a beginning or an end. So does that mean he cannot die? But could he cause himself to not exist anymore or does this go against the nature of God?
Allah is The Everlasting Alive الحيّ ALHAYY and He never dies as He Created the life and death on all His creatures and He is above whatever He Created.
We have no ability to know or even think about Him, but we should think about what He Created. The Hadeeth says: Don't think in Allah's Self but think in Allah's creatures.
Wassalam.