Hello, my name is Roman and I'm interested in your project.
If I understand correctly you have linphone app (for Android I guess?) and you have flexsip proxy. What you want to have is when flexsip passing call to some user it should send push notification and linphone app will up and ready to get incoming call. I have experience with Android/SIP. I never touched flexsip, but I see it has all required documentation so configuration or extending it should not be an issue.
Even if you would not avard my bid, we can just chat (it will cost nothing for you :)) about the topic and I can share how you can build this feature.
Feel free to contact me for discussion or any other questions.
Thanks,
Roman.