There's no evidence to suggest that the Roman and Han Chinese emperors ever made explicit contact with each other, however they did trade with each other. There was a long established trading sphere between the Mediterrenean, the Indian Ocean, and Asia in which Roman products would reach the far east, and Chinese products into Europe.

Some Roman sources briefly mention contact with the "Seres" (Latin for "silk people").

The Book of the Han (Hou-Han-Shu), a court document from the Han era, states;

“An-Tun, king of Daqin, sent an envoy…only then was contact established.”

An-Tun very likely means an Antonine emperor, probably Marcus Aurelius. And Daqin means "Greater Qin" = beyond China. It was an expression for all civilizations west of China.

So, contact between Rome and China never extended beyond trade and the occasional envoy. But they were certainly aware of each others' existence.