He had intercourse with his wife, thinking that dawn had not yet broken
Question

I had intercourse with my wife and I did not know that the adhaan for Fajr had already been given. I did not know that and I thought that the adhaan would be given a few minutes after five. After that I realized that the adhaan is given at quarter to five. What is the solution? Do I have to offer kafaarah, both me and my wife, knowing that we both did that willingly, and we had arrived from a journey 24 hours before and we did not yet know the timings of the prayer and we were taken by surprise when Ramadan was announced the morning after our arrival?.
Praise be to Allah.Firstly: 
If the matter is as you describe, then you do not have to do anything, because the one who does something that breaks the fast thinking that dawn has not yet broken, then he finds out that it had broken, does not have to make it up, according to the more correct of the two scholarly opinions, regardless of whether the thing that breaks the fast is food, drink or intercourse. 
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said: I have to point out that the things that break the fast, intercourse, food and drink, only break a person’s fast if three conditions are met:  
1 – That he knows about that. If he does not know then his fast is not broken, because Allah says (interpretation of the meaning):   
“And there is no sin on you concerning that in which you made a mistake,  except in regard to what your hearts deliberately intend. And Allah is Ever Oft‑Forgiving, Most Merciful” [al-Ahzaab 33:5] 
“Our Lord! Punish us not if we forget or fall into error”
[al-Baqarah 2:286] and Allah says: “I will do that.”
And the Prophet (peace and blessings of Allah be upon him) said: “My ummah is forgiven for what it does by mistake, forgets or is forced to do.” The one who is ignorant is making a mistake, even if he knows what he is doing. If he does something that breaks the fast out of ignorance, he does not have to do anything and his fast is complete and valid, whether he was ignorant of the ruling or of the time.  
An example of being ignorant of the ruling is if he consumes something that breaks the fast thinking that it does not break the fast, such as if he is treated with cupping, thinking that cupping does not break the fast. We say: Your fast is valid and you do not have to do anything. 
And example of being ignorant of the time is if he thinks that dawn has not yet broken, so he eats. His fast is valid. 
2 – That he remembers that he is fasting; if he forgets then it does not break the fast. 
3 – That he does it willingly. If he does not do it willingly then he does not break his fast. 
End quote from Majmoo’ Fatawa al-Shaykh Ibn ‘Uthaymeen (19/280). 
The Shaykh (may Allah have mercy on him) was asked: There is a man who is newly married and he had intercourse with his wife at the end of the night, thinking that it was still night, then the iqaamah was given. What do you think? Does he have to do anything? 
He replied: No, he does not have to do anything; there is no sin on him and he does not have to offer expiation or make up that day, because Allah, may He be exalted, says (interpretation of the meaning): 
“So now have sexual relations with them and seek that which Allah has ordained for you (offspring), and eat and drink until the white thread (light) of dawn appears to you distinct from the black thread (darkness of night)” al-Baqarah 2:187]
So these three things are all the same: intimacy with one’s wife, eating and drinking, and there is no evidence to suggest any distinction between them.  They are all things that are not allowed when fasting, and if they happen due to ignorance or forgetfulness, it does not matter. End quote from al-Liqa’ al-Shahri. 
Thus it is clear that you do not have to do anything, you do not have to make up that fast or offer expiation. This applies if you fasted on that day. But if you did not fast thinking that the fast had been invalidated because of intercourse, then all you have to do is make up that day. 
And Allah knows best.