Is there a similar function in postgresql for mysql's SQL_CALC_FOUND_ROWS?

No, there is not (at least not as of July 2007). I'm afraid you'll have to resort to:

BEGIN ISOLATION LEVEL SERIALIZABLE;

SELECT id, username, title, date FROM posts ORDER BY date DESC LIMIT 20;
SELECT count(id, username, title, date) AS total FROM posts;

END;

The isolation level needs to be SERIALIZABLE to ensure that the query does not see concurrent updates between the SELECT statements.

Another option you have, though, is to use a trigger to count rows as they're INSERTed or DELETEd. Suppose you have the following table:

CREATE TABLE posts (
    id      SERIAL PRIMARY KEY,
    poster  TEXT,
    title   TEXT,
    time    TIMESTAMPTZ DEFAULT now()
);

INSERT INTO posts (poster, title) VALUES ('Alice',   'Post 1');
INSERT INTO posts (poster, title) VALUES ('Bob',     'Post 2');
INSERT INTO posts (poster, title) VALUES ('Charlie', 'Post 3');

Then, perform the following to create a table called post_count that contains a running count of the number of rows in posts:

-- Don't let any new posts be added while we're setting up the counter.
BEGIN;
LOCK TABLE posts;

-- Create and initialize our post_count table.
SELECT count(*) INTO TABLE post_count FROM posts;

-- Create the trigger function.
CREATE FUNCTION post_added_or_removed() RETURNS TRIGGER AS $$
    BEGIN
        IF TG_OP = 'DELETE' THEN
            UPDATE post_count SET count = count - 1;
        ELSIF TG_OP = 'INSERT' THEN
            UPDATE post_count SET count = count + 1;
        END IF;
        RETURN NULL;
    END;
$$ LANGUAGE plpgsql;

-- Call the trigger function any time a row is inserted.
CREATE TRIGGER post_added_or_removed_tgr
    AFTER INSERT OR DELETE
    ON posts
    FOR EACH ROW
    EXECUTE PROCEDURE post_added_or_removed();

COMMIT;

Note that this maintains a running count of all of the rows in posts. To keep a running count of certain rows, you'll have to tweak it:

SELECT count(*) INTO TABLE post_count FROM posts WHERE poster <> 'Bob';

CREATE FUNCTION post_added_or_removed() RETURNS TRIGGER AS $$
    BEGIN
        -- The IF statements are nested because OR does not short circuit.
        IF TG_OP = 'DELETE' THEN
            IF OLD.poster <> 'Bob' THEN
                UPDATE post_count SET count = count - 1;
            END IF;
        ELSIF TG_OP = 'INSERT' THEN
            IF NEW.poster <> 'Bob' THEN
                UPDATE post_count SET count = count + 1;
            END IF;
        END IF;
        RETURN NULL;
    END;
$$ LANGUAGE plpgsql;