You are missing the update statement.
Create trigger [PMDB].[TG_count]
on [PMDB].[Alias_A] after insert, update
as
begin
if update(TableName) or update(PK) or update(FieldName) or exists (select * from deleted)
BEGIN
UPDATE a
set N_count = N_count + 1
FROM deleted d
INNER JOIN [PMDB].[Alias_A] a
ON a.PK = d.PK
END
end