Did Alexander the Great believe in the God of Abraham?

Answer 1

No. Alexander the great was a polytheist and worshiped the Greek pantheon of gods. However, he had no issue with the Jewish belief in the God of Abraham and explicitly permitted Jews to continue their faith when he conquered the Persian Empire. His successors, the Seleucids, were less tolerant than Alexander.

Additionally, there is a Related Link on Alexander's petition to the Jews not to resist the Macedonian Greek Army as it conquered the Persian Empire.

Answer 2

Yes - his own superiority and destiny to rule the known world. Having taken over the Persian Empire in the east, he was planning to take over Sicily and Carthage in the west when he died.
Alexander was also deeply religious and is recorded often paying respects and tributes to the Greek Gods. His religious belief and in himself also extended to imagining himself as Son of Zeus/Ammon after visiting Siwa Oasia and also being hailed as Son of Ra in Egypt. His Propaganda machine had also gained the Son of Zeus title from the oracles of Didyma and Erethrae in Greece.