You can insert new rows with modified values and then delete old rows. Following example change ID to be same as foreign key PersonId
SET IDENTITY_INSERT [PersonApiLogin] ON
INSERT INTO [PersonApiLogin](
[Id]
,[PersonId]
,[ApiId]
,[Hash]
,[Password]
,[SoftwareKey]
,[LoggedIn]
,[LastAccess])
SELECT [PersonId]
,[PersonId]
,[ApiId]
,[Hash]
,[Password]
,[SoftwareKey]
,[LoggedIn]
,[LastAccess]
FROM [db304].[dbo].[PersonApiLogin]
GO
DELETE FROM [PersonApiLogin]
WHERE [PersonId] <> ID
GO
SET IDENTITY_INSERT [PersonApiLogin] OFF
GO