In MySQL, I have two different databases -- let's call them A and B.
Is it possible to perform a join between a table that is in database A, to a table that is in database B?
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.