He is asking about Masjid al-Qiblatayn and the reason for its name

This incident happened in more than one of the mosques of Madinah at that time, yet despite that the mosques were not called by this name.

The Sahabah (may Allah be pleased with them) turned around to face towards the Ka‘bah whilst they were praying in Masjid Quba’, as is proven in the hadith of Ibn ‘Umar (may Allah be pleased with him).

They also turned around whilst they were praying in the masjid of Banu Harithah, as was mentioned in the hadith of al-Bara’ ibn ‘Azib (may Allah be pleased with him).

What was mentioned by the biographers about Masjid al-Qiblatayn and the reason for its name has to do with the masjid of Banu Salamah. It was said that the Prophet (blessings and peace of Allah be upon him) visited Umm Bishr, and the time came for Zuhr prayer. Whilst he was praying, he was commanded to face towards the Ka‘bah, so he turned around. The scholars and commentators attribute this report to Tabaqat Ibn Sa‘d.

Ibn Hajar (may Allah have mercy on him) said:

The reports differ concerning the prayer during which the qiblah was changed, and the mosque in which that happened…

Muhammad ibn Sa‘d said in al-Tabaqat: It was said that he had led the Muslims in praying two rak‘ahs of Zuhr in his mosque, then he was commanded to face towards al-Masjid al-Haram, so he turned around to face it and the Muslims turned around with him.

It was also said that the Prophet (blessings and peace of Allah be upon him) visited Umm Bishr ibn al-Bara’ ibn Ma‘rur in Bani Salamah, and she made some food for him. The time for Zuhr came, so the Messenger of Allah (blessings and peace of Allah be upon him) led his companions in praying two rak‘ahs, then he was commanded to turn around to face the Ka‘bah, so he turned to face towards it. Hence that place was named Masjid al-Qiblatayn.

Ibn Sa‘d said: al-Waqidi said: This is more sound in our view."(Fath al-Bari  2/116).

This was also narrated in al-Tabaqat al-Kubra (1/241).

Ibn Kathir said:

More than one of the commentators and others stated that the change of the qiblah was revealed to the Messenger of Allah (blessings and peace of Allah be upon him) when he had prayed two rak‘ahs of Zuhr. That happened in the mosque of Banu Salamah, which became known as Masjid al-Qiblatayn.

In the hadith of Nuwaylah bint Muslim, it says that news of that reached them when they were praying Zuhr, so the men moved to the women’s place and the women moved to the men’s place. This was quoted by Shaykh Abu ‘Umar ibn ‘Abd al-Barr al-Nimri.

With regard to the people of Quba’, the news did not reach them until the time of Fajr prayer on the following day, as it was narrated in al-Sahihayn from Ibn ‘Umar (may Allah be pleased with them both) that he said: Whilst the people in Quba’ were praying Fajr, someone came to them and said: ‘Last night Qur’an came down to the Messenger of Allah (blessings and peace of Allah be upon him) and he was instructed to face towards the Ka‘bah, so turn around to face towards it.’ They were facing towards al -Sham (Greater Syria), then they turned around to face towards the Ka‘bah."(Tafsir Ibn Kathir  1/237).

This hadith – as is clear – was quoted by Ibn Sa‘d without any isnad, and he quoted it with wording which indicates that there is some doubt and that he was not certain of its soundness, as he said: “It was said… it was said.” Such a report cannot be relied on and there can be no certainty that it is sound.

But these are historical reports and no action is based on them. But there is nothing wrong with narrating them by way of telling the story, not by way of affirming its soundness.

It is not recommended for the visitor to Madinah to go and visit any of these places in Madinah and its environs, whether the hadith about the Prophet (blessings and peace of Allah be upon him) praying there is sound or not.

Please see also the answer to question no. 11669 .

And Allah knows best.