What the imam is doing before bowing in the second rak’ah is Qunoot, and saying it silently is permissible according to the Maaliki madhhab. They regarded it as mustahabb to make Qunoot in Fajr prayer silent. This is also one of the two views of the Shaafa’is. 

The basic principle concerning Qunoot in Fajr is the subject of some difference of scholarly opinion. Some of them, such as the Maalikis and Shaafa’is, think that it is prescribed, and some of them, such as the Hanafis and Hanbalis, do not think that this is the case. 

Ibn Qudaamah (may Allaah have mercy on him) said in al-Mughni (1/449): It is not Sunnah to say Qunoot in Fajr prayer or any other prayer, apart from Witr. This is the view of al-Thawri and Abu Haneefah, and it was narrated from Ibn ‘Abbaas, Ibn ‘Umar, Ibn Mas’ood and Abu’l-Darda’. 

Maalik, Ibn Abi Layla, al-Hasan ibn Saalih and al-Shaafa’i said that it is Sunnah to say Qunoot in Fajr at all times, because Anas said: The Messenger of Allaah (blessings and peace of Allaah be upon him) said Qunoot in Fajr until he departed this world. This was narrated by Imam Ahmad in al-Musnad. ‘Umar said Qunoot in Fajr in the presence of the Companions and others. 

And we have what is narrated, that the Prophet (blessings and peace of Allaah be upon him) said Qunoot for a month, praying against one of the tribes of the Arabs, then he stopped doing that. Narrated by Muslim. And Abu Hurayrah and Abu Mas’ood narrated something similar from the Prophet (blessings and peace of Allaah be upon him). And it was narrated that Abu Maalik said: I said to my father: O my father, you prayed behind the Messenger of Allaah (blessings and peace of Allaah be upon him), and behind Abu Bakr, ‘Umar and ‘Uthmaan, and behind ‘Ali here in Kufah for about five years. Did they say Qunoot? He said: That is an innovation. Al-Tirmidhi said: this is a saheeh hasan hadeeth, and should be followed according to most of the scholars. Ibraaheem al-Nakha’i said: The first one who said Qunoot in Fajr prayer was ‘Ali, because he was a man at war who prayed against his enemies. Sa’eed narrated in his Sunan from Hushaym, from ‘Urwah al-Hamadhaani, that al-Shu’bi said: When ‘Ali said Qunoot in Fajr prayer, the people objected. ‘Ali said: We are only asking Allaah for support against our enemies. And it was narrated that Abu Hurayrah (may Allaah be pleased with him) said: The Messenger of Allaah (blessings and peace of Allaah be upon him) did not say Qunoot in Fajr prayer except when he was praying for some people or praying against some people. Narrated by Sa’eed. The hadeeth of Anas may be understood as meaning that he stood for a long time, because that may be called Qunoot. And the Qunoot of ‘Umar may be understood as having happened at times of calamity, because most of the reports from him indicate that he did not say Qunoot, but a number of people narrated that from him, which indicates that his Qunoot was only in times of calamity. End quote. 

See: al-Mawsoo’ah al-Fiqhiyyah, 34/58 

Although the more correct view is that there should be no Qunoot in Fajr except in times of calamity, there is nothing wrong with praying behind someone who says Qunoot in Fajr and saying Ameen to his du’aa’. 

The Shaafa’is say that it is permissible to pray Zuhr and ‘Asr behind someone who is praying Fajr and Maghrib, and they said that it does not matter if one follows the imam in Qunoot when he is praying Fajr, and to sit for the second tashahhud when the imam is praying Maghrib, as in the case of one who joins the prayer late, and he may stop following the imam, but following the imam is better than stopping following him.

If it is said: how can it be permissible for the one who is praying behind an imam to follow the imam in Qunoot even though it is not prescribed for the one who is praying behind him, my answer is that this is forgiven for the sake of following the imam. Mughni al-Muhtaaj by al-Sharabeeni, 1/245 

Shaykh al-Islam Ibn Taymiyah (may Allaah have mercy on him) said: 

If the person praying behind an imam is following someone who says Qunoot in Fajr or Witr, he should say Qunoot with him, whether he says Qunoot before or after bowing, but if he does not usually say Qunoot he should not say Qunoot with him. 

If the imam thinks that something is mustahabb and the people praying behind him do not think that it is mustahabb, and he refrains from it the sake of agreement and harmony, then he has done well. End quote. Majmoo’ al-Fataawa, 22/268. 

See also question no. 20031, 5459, 59925. 

With regard to what we have mentioned about following the imam, it makes no difference whether the imam says Qunoot out loud or silently. If the imam says it out loud, then those who are praying behind him should say Ameen to his du’aa’, and if he says it silently, as mentioned in the question, then the person praying behind him should say Qunoot to himself, until the imam has finished. 

Ibn Muflih quoted in al-Furoo’, 1/542, a report from Imam Ahmad that if he does not hear the imam, he should say du’aa’, even if this is in the Qunoot of Witr. It is stated that with regard to the one who is praying behind an imam who says Qunoot in Fajr, there are two reports: one says that he should keep quiet and the other says that he should follow him as in Witr. Al-Mardaawi said in his Tasheeh: The correct view is that he should follow him, so he should say Ameen and offer du’aa’. End quote. 

Hence Shaykh Ibn Qaasim (may Allaah have mercy on him) stated in Haashiyat al-Rawd that he should follow him here, and he commented on the words in al-Rawd, “Whoever follows an imam who says Qunoot in Fajr should follow him and say Ameen” by noting (2/199):

i.e., he should follow the imam in his du’aa’, because of the hadeeth: “The imam has only been appointed to be followed, so do not differ from him.” Similarly, the person who is praying behind an imam should say Ameen to his imam’s du’aa’ if he hears Qunoot, and if he does not hear it he should say du’aa’. End quote. 

And Allaah knows best.