How to update only part of a field
You can try something like this:

UPDATE Set name = REPLACE(t.name, r.bad, r.good)
FROM yourTable t
INNER JOIN (values
    ('FY15-Q2', 'FY16-Q1')
    , ('FY25-Q2', 'FY36-Q1')
    , ...
) as r(bad, good)
    On t.name like '%'+bad


You can add multiple pattern after FY25-Q2.

It should work fine if you don't have multiple matches. It may not be really efficient if you have tons of rows.

If your pattern is in the middle of the string, add a % at the end as well: On t.name like '%'+bad+'%'.

Before running the full update, you should first replace UPDATE by SELECT and look at what should be done:

SELECT t.name, newname = REPLACE(t.name, r.bad, r.good), r.bad, r.good
...