Praise be to Allah.
If you were unaware of the actual time when Fajr begins in the city to which you have moved, and you ate after the time for Fajr began, and you did that by mistake, then there is a difference of scholarly opinion about the one who eats or drinks, thinking that it is still night and that dawn has not broken, and also about the one who eats or drinks thinking that the sun has set, then finds out that he was wrong.
Many of the scholars are of the view that the person’s fast is invalidated by that, and he must fast a day to make up for that.
But some of the scholars are of the view that his fast is still valid, and that he should complete his fast, and he does not have to make it up.
This is the view of Mujaahid and al-Hasan among the Taabi‘een, and is mentioned in one report narrated from Imam Ahmad. It was also the view favoured by al-Muzani among the Shaafa‘is, and by Shaykh al-Islam Ibn Taymiyyah. It was also the view deemed more likely to be correct by Shaykh Muhammad as-Saalih al-‘Uthaymeen (may Allah be pleased with him).
It was narrated that Sahl ibn Sa‘d said: The words “And eat and drink until the white thread becomes distinct to you from the black thread” [al-Baqarah 2:187] were revealed, and the words “of dawn” were not revealed, so if people wanted to fast, one of them would tie a white thread to one foot and a black thread to the other, then he would carry on eating and drinking until he could tell them apart when he saw them. Then after that Allah revealed the words, “of dawn,” then they realized that what was meant by that was night and day. Narrated by al-Bukhaari (1917) and Muslim (1091).
Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said:
Whatever a person omits of an obligatory duty because he is unaware of it, such as one who prays hurriedly and without being at ease in the prayer, because he does not know that being at ease in the prayer is obligatory, then in this case there is a difference of scholarly opinion: should he repeat it after the time for the prayer has ended or not? There are two well-known views, both of which are mentioned in the madhhab of Ahmad and others.
The correct view is that in such cases, he does not have to repeat the prayer, because it is proven from the Prophet (blessings and peace of Allah be upon him) in as-Saheeh that he said to the Bedouin who prayed badly: “Go and pray, for you have not prayed” – two or three times – then he said: By the One Who sent you with the truth, I cannot do any better than this, so teach me what is required of me in my prayer. So the Prophet (blessings and peace of Allah be upon him) taught him to how to pray unhurriedly and be at ease in the prayer, and he did not instruct him to repeat the prayers he had done before that time, even though the man said: By the One Who sent you with the truth, I cannot do any better than this. Rather he instructed him to repeat that prayer because the time for it had not yet ended, and he was enjoined to offer that prayer during its time. As for the prayers for which the times had ended, he did not instruct him to repeat them, even though he had omitted some of their essential parts, because he had not been aware that that was obligatory upon him.…
Similarly, in the case of those who ate during Ramadan until one of them could distinguish between the white thread and the black thread, and they ate until after dawn had broken, the Prophet (blessings and peace of Allah be upon him) did not instruct them to repeat those fasts. Those people were unaware of what was obligatory, so he did not instruct them to make up what they had omitted when they were unaware of the matter. That is similar to the case of a disbeliever who becomes Muslim: he is not to be instructed to make up what he did not do at the time when he was a disbeliever and unaware of Islamic teachings.
End quote from Majmoo‘ al-Fataawa (21/429-431).
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said:
Ignorance – may Allah bless you – means not being aware of something, but sometimes a person may be excused for being unaware of some issues of the past, but not of the present. One example of that is what is narrated in as-Saheehayn from Abu Hurayrah, that a man came and offered a prayer in which he hurried and was not at ease in it, then he came and greeted the Prophet (blessings and peace of Allah be upon him) with salaam, and he told him: “Go back and pray, for you have not prayed.” That happened three times, then the man said to him: By the One Who sent you with the truth, I cannot do any better than that, so teach me. So he taught him, but he did not instruct him to make up his past prayers, because he was unaware before. Rather he instructed him to repeat the prayer that was currently due.
End quote from Liqa’ al-Baab al-Maftooh (19/32).
For more information, please see the answer to question no. 38543.
To sum up: you are excused because you were unaware of the different time of Fajr in your new city, and your fasts are valid.
Something similar to this happened to some of the Sahaabah (may Allah be pleased with them), and there is no report that says that the Prophet (blessings and peace of Allah be upon him) instructed them to make up the fasts.
Nevertheless, if you err on the side of caution and repeat the making up of those days, that will be better and furthest removed from uncertainty, and it will avoid a matter concerning which there is a scholarly difference of opinion.
And Allah knows best.