Join between tables in two different databases?

Yes, assuming the account has appropriate permissions you can use:

SELECT <...>
FROM A.table1 t1 JOIN B.table2 t2 ON t2.column2 = t1.column1;

You just need to prefix the table reference with the name of the database it resides in.

SELECT <...> 
FROM A.tableA JOIN B.tableB 

SELECT *
FROM A.tableA JOIN B.tableB 

or

SELECT *
  FROM A.tableA JOIN B.tableB
  ON A.tableA.id = B.tableB.a_id;

SELECT <...>
FROM A.table1 t1 JOIN B.table2 t2 ON t2.column2 = t1.column1;

Just make sure that in the SELECT line you specify which table columns you are using, either by full reference, or by alias. Any of the following will work:

SELECT *
SELECT t1.*,t2.column2
SELECT A.table1.column1, t2.*
etc.