The answer is yes, you can.

Try it like this:

UPDATE TABLE_A a
    JOIN TABLE_B b ON a.join_col = b.join_col AND a.column_a = b.column_b
    JOIN TABLE_C c ON [condition]
SET a.column_c = a.column_c + 1

For a general update join:

UPDATE TABLEA a
JOIN TABLEB b ON a.join_colA = b.join_colB
SET a.columnToUpdate = [something]