Is there any relation between the Cyrillic Ш (sha) and the Chinese 山 (shān)?
No. The Cyrillic script is based on the Greek script, and some other local scripts like Hebrew. The basis for sha is thought to be the Hebrew letter ש (shin).

ש

It's unlikely that shin is based on the Chinese character, either. It seems taken from the Phoenician alphabet, where the corresponding letter looks like a Latin W.
It's worth keeping in mind that at the time Cyrillic characters were invented, Russia did not yet control Siberia. So it was very far from China, and contact between China and the west was limited.

But maybe the Chinese character was influenced by these scripts?
It's not impossible, but the earliest versions of the Character look more like a triangle, so the simpler explanation is that it's just a picture of a mountain:



Considering the evidence, it seems unlikely that these characters are related.