No, not like this. But you can raise and maintain your own exceptions, no problem:
CREATE TABLE exceptions(
    id serial primary key,
    MESSAGE text, 
    DETAIL text, 
    HINT text, 
    ERRCODE text
);
INSERT INTO exceptions (message, detail, hint, errcode) VALUES ('wrong', 'really wrong!', 'fix this problem', 'P0000');
CREATE OR REPLACE FUNCTION foo() RETURNS int LANGUAGE plpgsql AS
$$
DECLARE
    row record;
BEGIN
    PERFORM * FROM fox; -- does not exist, undefined_table, fail
    EXCEPTION
        WHEN undefined_table THEN
            SELECT * INTO row FROM exceptions WHERE id = 1; -- get your exception
            RAISE EXCEPTION USING MESSAGE = row.message, DETAIL = row.detail, HINT = row.hint, ERRCODE = row.errcode;
    RETURN 1;
END;
$$
SELECT foo();
Offcourse you can also have them hardcoded in your procedures, that's up to you.