Yes, but you need to move the variable assignment into the query:
SET @user := 123456;
SELECT @group := `group` FROM user WHERE user = @user;
SELECT * FROM user WHERE `group` = @group;
Test case:
CREATE TABLE user (`user` int, `group` int);
INSERT INTO user VALUES (123456, 5);
INSERT INTO user VALUES (111111, 5);
Result:
SET @user := 123456;
SELECT @group := `group` FROM user WHERE user = @user;
SELECT * FROM user WHERE `group` = @group;
+--------+-------+
| user | group |
+--------+-------+
| 123456 | 5 |
| 111111 | 5 |
+--------+-------+
2 rows in set (0.00 sec)
Note that for SET, either = or := can be used as the assignment operator. However inside other statements, the assignment operator must be := and not = because = is treated as a comparison operator in non-SET statements.
UPDATE:
Further to comments below, you may also do the following:
SET @user := 123456;
SELECT `group` FROM user LIMIT 1 INTO @group;
SELECT * FROM user WHERE `group` = @group;