What is the supplication that the Prophet (blessings and peace of Allah be upon him) learned from Allah, may He be exalted?
Question

Is there any such thing as the supplication that the Messenger (blessings and peace of Allah be upon him) said and that his Lord taught him?
Praise be to Allah.If you mean a specific supplication, we do not know what this supplication is, so that we could verify its soundness.
But if you do not mean a specific supplication, and you are looking for any supplication that meets this description, then the basic principle is that all the supplications that the Prophet (blessings and peace of Allah be upon him) taught us are revelation from Allah, may He be exalted, to him.
The Prophet’s Sunnah is revelation from Allah, may He be exalted.
Allah, may He be exalted, says (interpretation of the meaning):
{Your companion [Muhammad] has not strayed, nor has he erred,
Nor does he speak from [his own] inclination.
It is not but a revelation revealed} [an-Najm 53:2-4].
Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said:
… The Messenger of Allah (blessings and peace of Allah be upon him) said: “Indeed I have been given the Book and something like it” – meaning the Sunnah.
The Sunnah also came down to him by revelation, as the Qur’an came down, but it is not recited as the Qur’an is recited. Imam ash-Shafa‘i and other leading scholars quoted a great deal of evidence to that effect."(Majmu‘ al-Fatawa 13/363-364).
We are enjoined by Allah, may He be exalted, to follow the example of the Prophet (blessings and peace of Allah be upon him) in offering supplications.
Allah, may He be exalted, says (interpretation of the meaning):
{There has certainly been for you in the Messenger of Allah an excellent pattern for anyone whose hope is in Allah and the Last Day and [who] remembers Allah often} [al-Ahzab 33:21].
Ibn Kathir (may Allah have mercy on him) said:
This verse offers great proof that we must follow the example of the Messenger of Allah (blessings and peace of Allah be upon him) in his words and deeds and character…
{There has certainly been for you in the Messenger of Allah an excellent pattern} means: why don’t you emulate him and follow his example? Hence Allah says: {for anyone whose hope is in Allah and the Last Day and [who] remembers Allah often}."(Tafsir Ibn Kathir  6/391).
Conclusion: all the proven prophetic supplications meet the description that you asked about.
And Allah knows best.