Ways to "query a query" include:
Create a view. Use a derived table, aka a subquery with an alias. Here is how that one works:
select count(distinct alert_recipient, id)
from (SELECT alert.alert_recipient,alert_sent.id
FROM alert
INNER JOIN alert_sent
ON alert_sent.alert_id=alert.id ) temp
order id desc
limit 10
Or, you can create a text file with this sql.
SELECT alert.alert_recipient,alert_sent.id
FROM alert
INNER JOIN alert_sent
ON alert_sent.alert_id=alert.id
order id desc
limit 10
and copy and paste it into your query tool whenever you want to run it. Or, some query tools will allow you to save query files so you can run them whenever you want.