Praise be to Allah.
Lying down after praying two rak’ahs of Fajr is Sunnah, because of the report narrated by al-Bukhaari (626) from ‘Aa’ishah who said: “When the muezzin fell silent after the first call to Fajr prayer, the Messenger of Allaah (peace and blessings of Allaah be upon him) would get up and pray two short rak’ahs before Fajr prayer, after the dawn had broken, then he would lie down on his right side until the muezzin came to him to recite the iqaamah.”
Muslim narrated a more complete version: “Between the end of ‘Isha’ prayer – which the people call al-‘Atamah (darkness prayer) – and Fajr, the Messenger of Allaah (peace and blessings of Allaah be upon him) would pray eleven rak’ahs, saying the salaam after each two rak’ahs, and praying one rak’ah on its own as Witr. When the muezzin fell silent after calling to Fajr prayer, and he could see the dawn, and the muezzin came to him, he would get up and pray two short rak’ahs, then he would lie down on his right side until the muezzin came to him to recite the iqaamah.”
According to some reports, he lay down before the adhaan for Fajr. Al-Nawawi (may Allaah have mercy on him) said in his commentary on Muslim:
What is correct is that he lay down after the Sunnah of Fajr, because of the hadeeth of Abu Hurayrah who said: The Messenger of Allaah (peace and blessings of Allaah be upon him) said: “When any one of you has prayed two rak’ahs of Fajr, let him lie down on his right side.” (Narrated by Abu Dawood and al-Tirmidhi with an isnaad which is saheeh according to the conditions of al-Bukhaari and Muslim. Al-Tirmidhi said it is a saheeh hasan hadeeth). This saheeh hadeeth is clear about the matter of his lying down. With regard to the hadeeth of ‘Aa’ishah which says that he lay down before and after Fajr, and the hadeeth of Ibn ‘Abbaas which says that he lay down before, there is no contradiction, because the fact that he lay down before does not mean that he did not lie down afterwards as well. There are some reports which say that on some occasions he (peace and blessings of Allaah be upon him) did not lie down afterwards, which could be taken to demonstrate that it is permissible not to do it. That would apply if it were proven that he did not do it but it was not proven. Perhaps he used to lie down both before and after. If the ahaadeeth say that he told them to lie down after praying and that he himself did that, this proves the point. If it is possible to reconcile the ahaadeeth, it is not permissible to reject any of them, and it is possible to reconcile them in two ways, one of which we have mentioned, which is that he lay down both before and afterwards. The second is that he did not lie down afterwards on some occasions, to demonstrate that it is permissible not to do so. And Allaah knows best.
Some of the salaf were of the view that this is permissible at home but not in the mosque. This was narrated from Ibn ‘Umar, and was regarded as a strong opinion by some of our shaykhs, because it is not narrated that the Prophet (peace and blessings of Allaah be upon him) did this in the mosque. It was narrated that Ibn ‘Umar used to throw pebbles at those who did this in the mosque; this was narrated by Ibn Abi Shaybah.
Shaykh Ibn ‘Uthaymeen said, after discussing the differences of scholarly opinion concerning this lying down:
The correct view is that which was expressed by Shaykh al-Islam [Ibn Taymiyah], that if a person is tired after his tahajjud (prayers at night), he may rest by lying down on his right side. This is on condition that there is no fear of his being overtaken by sleep that will make him miss the prayer. If there is such a fear then he should not sleep.
Sharh Riyaadh al-Saaliheen, 3/287
The reason for this lying down is to rest if one is tired after praying qiyaam al-layl (naafil prayers at night), so that one's energy will be renewed for Fajr prayer. Based on this, what some people do by lying down for less than a minute, as mentioned in the question, do not achieve the purpose, and it goes against the Sunnah because the Prophet (peace and blessings of Allaah be upon him) used to lie down until the muezzin came to him to say the iqaamah just before the prayer.
And Allaah knows best.