No, they are two different mosques, in two different directions.
You can see Masjid Qiblatain marked near the top left as al-Qiblatain, while Masjid Quba is towards the bottom of the picture, near where it says Kuba.
They look similar, but Masjid Quba was the one made upon arrival in Madina, but the one the Prophet :saws: was praying in at the time of the qibla change became known is Masjid Qiblatain.
Masjid Quba has special virtues attached with visiting it, which Masjid Qiblatain does not, though many do visit Masjid Qiblatain out of interest.
These are some of the ahadeeth regarding the virtues of praying in Masjid Quba:
It is prescribed for the visitor to Madeenah and for the one who lives there to go to the Mosque of Quba’ and pray there, following the example of the Prophet (peace and blessings of Allaah be upon him) and seeking the reward of ‘Umrah. It was narrated that Sahl ibn Haneef said: The Messenger of Allaah (peace and blessings of Allaah be upon him) said: “Whoever goes out until he comes to this mosque – meaning the Mosque of Quba’ – and prays there, that will be equivalent to ‘Umrah.” Narrated by Ahmad, 3/437; al-Nasaa’i, 699; classed as saheeh by al-Albaani in Saheeh al-Targheeb, 1180, 1181.
According to Ibn Maajah: “Whoever purifies himself in his house, then comes to the Mosque of Quba’ and prays there, he will have the reward of ‘Umrah.” Narrated by Ibn Maajah, 1412.
In Sahih Muslim it is narrated that the Messenger of Allaah (peace and blessings of Allaah be upon him) used to go to the mosque of Quba’ every Saturday, walking or riding, and he would pray two rak’ahs there. Narrated by al-Bukhaari, 1191; Muslim, 1399
[Taken from Etiquettes of Ziyaarah - by Imâm of Masjid al Nabawi, Sheikh Salâh al-Budayr ]
None of these ahadith apply to Masjid Qiblatain.
Masjid Qiblatain, which has only two minarets: