Replace Character Without Using Looped REPLACE Function

Create a mapping between the old value and the new value in a table value constructor and use stuff to replace the character.

declare @T table(Value varchar(10));

insert into @T values
('Q234567888'),
('R264747848'),
('B264712481');

select stuff(T.Value, 1, 1, (
                            select V.NewValue
                            from (values('Q', 'I'),
                                        ('R', 'I'),
                                        ('B', 'U')) as V(OldValue, NewValue)
                            where V.OldValue = left(T.Value, 1)
                            ))
from @T as T;

DECLARE @values TABLE
 (  
     String CHAR(10)
 )

INSERT INTO @values
values
('Q234567888'),
('R264747848'),
('B264712481');

SELECT 
    REPLACE(String,LEFT(String,1),ca.NewValue)
FROM @values
CROSS APPLY 
(SELECT * FROM (values('Q', 'I'),
('R', 'I'),
('B', 'U'))as V(OldValue, NewValue)
WHERE v.OldValue = LEFT(string,1)                               
)ca

Here's another approach. While testing this, I ran @Mikael Eriksson's method and the CASE\REPLACE with 26 clauses, over a 3 million row test table and the CASE method was quickest hands down. That said, none were slow and the above and @Mikael's shook out evenly. Of course, YMMV.

Just to have this stated as an option since the O.P.'s version of SQL Server has not been mentioned, and the following is a valid approach starting with SQL Server 2017:

The TRANSLATE function is well suited for a direct 1-to-1 replacement of single characters:

DECLARE @TEST TABLE([Stuff] VARCHAR(10));

INSERT INTO @TEST ([Stuff]) VALUES
('Q234567888'),
('R264747848'),
('B264712481');

UPDATE tst
SET    tst.[Stuff] = 
          STUFF(tst.[Stuff], 1, 1, (
              TRANSLATE(LEFT(tst.[Stuff], 1), 'QRB', 'IIU')
              ))
FROM   @TEST tst;

SELECT * FROM @TEST;

returns:

Stuff
----------
I234567888
I264747848
U264712481

P.S. Ignore SSMS (at least through v 17.5, I haven't yet tried 17.6) flagging TRANSLATE as an invalid function. This behavior has been reported here.