Postgres on conflict do update on composite primary keys

Just place both keys in the ON CONFLICT clause:

INSERT INTO answer VALUES (1,1,'q1') 
ON CONFLICT (person_id,question_id) 
DO UPDATE SET answer = EXCLUDED.answer; 

Example:

INSERT INTO answer VALUES (1,1,'q1') 
ON CONFLICT (person_id,question_id) 
DO UPDATE SET answer = EXCLUDED.answer;             

SELECT * FROM answer;
 person_id | question_id | answer 
-----------+-------------+--------
         1 |           1 | q1
(1 Zeile)

INSERT INTO answer VALUES (1,1,'q1') 
ON CONFLICT (person_id,question_id) 
DO UPDATE SET answer = EXCLUDED.answer || '-UPDATED';             

SELECT * FROM answer;
 person_id | question_id |   answer   
-----------+-------------+------------
         1 |           1 | q1-UPDATED
(1 Zeile)

Demo: db<>fiddle

PostgreSQL 15 +

You can also achieve the same result using MERGE:

MERGE INTO answer i
  -- records to be inserted
  USING (
    VALUES (1,1,'q1'), -- already exists in table answers!
           (2,2,'q2')  -- new record
  ) AS j (person_id, question_id, answer)
  -- checking if the PK of given records (j) already exists
  -- in table "answer" (i).
  ON j.question_id = i.question_id AND j.person_id = i.person_id
WHEN MATCHED THEN
  -- in case of a match (conflict), I want to add the suffix '-UPDATED' 
  -- to the column "answer"
  UPDATE SET answer = j.answer || '-UPDATED'
WHEN NOT MATCHED THEN 
  -- if there is no match (conflict) just INSERT the record.
  INSERT (person_id, question_id, answer)
  VALUES (j.person_id, j.question_id, j.answer);

Demo: db<>fiddle