Can I notify Fili first and then Kili only if Fili does not acknowledge the alert in 10 minutes?

Yes. When usernames and/or groups specified as the recipients, all specified users are notified immediately. To notify users in order with a time delay OpsGenie provides "escalations". In an escalation, OpsGenie tries each user and/or group in order until the alert is acknowledged by someone (or closed).

To notify first Fili, and then Kili, first create an escalation, and specify the names of the users and/or groups and the time delay between each in the escalation rules.

Then when creating the alert, simply put the name of the escalation into the recipients field.