In To Kill a Mockingbird by Harper Lee, can Tom Robinson be Calpurnia's long forgotten brother since the reader never learns her last name?

It is unlikely that Tom Robinson is related to Calpurnia. 

First, it seems likely that a blood relationship between Tom and Calpurnia would come up in the course of the novel. Calpurnia is involved in Tom's case through Atticus and through her own involvement in Maycomb's black community—if the two were related, it would be unusual not to encounter mention of it somewhere in the story.

Also, Calpurnia is considerably older than Tom, possibly old...

It is unlikely that Tom Robinson is related to Calpurnia. 

First, it seems likely that a blood relationship between Tom and Calpurnia would come up in the course of the novel. Calpurnia is involved in Tom's case through Atticus and through her own involvement in Maycomb's black community—if the two were related, it would be unusual not to encounter mention of it somewhere in the story.

Also, Calpurnia is considerably older than Tom, possibly old enough to be Tom's mother. After all, Tom's children are younger than Zeebo's. As Scout narrates in Chapter 12,

Zeebo was Calpurnia's eldest son. If I had ever thought about it, I would have known that Calpurnia was of mature years--Zeebo had half-grown children--but then I had never thought about it.

Another thing about Calpurnia that sets her apart as not a relative of anyone in the Maycomb community is the fact that Cal does not speak as the others do.