When I run the following query in SQL Server 2019, the result is 1, whereas it should be 0.
select CHARINDEX('αρ', 'αυρ')
What could be the problem?
As was mentioned in the comments it may be because you have not declared your string literals as Unicode strings but are using Unicode characters in the strings. SQL Server will be converting the strings to another codepage and doing a bad job of it. Try running this query to see the difference.
SELECT 'αρ', 'αυρ', N'αρ', N'αυρ'
On my server, this gives the following output:
a? a?? αρ αυρ
Another issue is that CHARINDEDX uses the collation of the input which I think is probably not set correctly in this instance. You can force a collation by setting it on one of the inputs. It is also possible to set it at the instance, database and column level.
There are different collations that may be applicable. These have different features, for example some are case sensitive some are not. Also not all collations are installed with every SQL Server instance. It would be worth running SELECT * from sys.fn_helpcollations() to see the descriptions of all the installed ones.
If you change your query to this you should get the result you are looking for.
SELECT CHARINDEX(N'αρ' COLLATE Greek_BIN, N'αυρ')