Both teach nonattachment, impermanence, and interconnectedness.
Both advise self-control, especially when strong emotions are involved.
Both teach that how we think about things determines how we experience life.
Both say that we create our own suffering by constantly yearning for more while failing to appreciate what we have.
And most of all, both place a strong emphasis on virtuous thoughts and actions.
But there are differences as well. Stoicism focuses on reason rather than mysticism. Concepts like Nirvana and rebirth are absent from Stoicism, as is the Buddhist practice of meditation.
The historical record is scant. I created this crude timeline to show the key interactions between Greek and Buddhist cultures:
You’ll notice that there is no known interactions between Buddhists and Stoics in ancient times. Stoicism grew out of Cynicism, however, and Cynic philosopher Onesicritus did interact with Indian ascetics after Alexander the Great reached the Indus River. We don’t know if these Indian ascetics were Buddhist, though they could have been. Besides, Cynicism had independently developed asceticism and non-attachment prior to contact with the East.
After Alexander’s empire split into smaller empires, Indo-Greek King Menander I became a Buddhist. And through trade routes it’s possible that some Buddhist ideas made their way back to Greece. And Caesar Augustus is known to have met with a Buddhist Indian king. A century and a half later the Stoic philosophy of Emperor Marcus Aurelius emphasized impermanence and interconnectedness.
But this is all circumstantial. It’s possible that Buddhist thought had an indirect influence on Stoicism, first through Cynic contact with the East and later through trades routes. But if so, it’s unlikely that Stoic philosophers knew the Buddhist origin of these influences.