Apparently not - at least not recent descent. According to the Harper Encyclopedia of Military Biography (Dupuy, Johnson, and Bongard) he was "born March 25, 1767 at La Bastide-Fortunière (now Labastide-Murat), Quercy, in Gascony, the son of Pierre Murate Jordy, a farmer and innkeeper".

Since this book focuses on the military careers of its subjects it does not offer any more on his ancestors, other than to imply that they were not military since it always seems to mention those cases. There is over a page dedicated to him here closing with a quote from Napoleon : "The bravest of men in the face of the enemy, incomparable on the battlefield, but a fool in his actions everywhere else."

A valid premise for research might be that simply "Murat" is looking and perhaps even sounding like a word or name of Turkish/Arabic origin, therefore Joachim Murat may have a family name of similar decent.

One problem not accounted for is that this might be a false premise, since it was apparently left unchecked whether "murat" might be actually something inherently French.

And Murat is actually found several times in France as a toponym or hydronym:

French Wikipedia: Murat

Toponymes

Murat est un nom de lieu notamment porté par :

Murat, commune française de l'Allier
Murat, commune française du Cantal
Murat (commune déléguée), ancienne commune française du Cantal, commune déléguée intégrée à Murat (Cantal)
Murat-la-Rabe, ancien fief avec château sur la commune française de la Monselie dans le Cantal
Murat-le-Quaire, commune française du Puy-de-Dôme
Murat-Orsennes, ancienne commune française de l'Indre
Murat-sur-Vèbre, commune française du Tarn
Murat, lieu-dit près d'Espalion (Aveyron) d'où est originaire la famille de Murat de L'Estang
Bourg-Murat, village de La Réunion
Chastel-sur-Murat, ancienne commune française du Cantal
Gourdon-Murat, commune française de la Corrèze
Labastide-Murat, commune française du Lot
Saint-Marcel-en-Murat, commune française de l'Allier
Saint-Priest-en-Murat, commune française de l'Allier
Murat, Lieu-dit appartenant à la commume de Cazillac

Hydronyme

Special mention for this has to go to the old viscounties of Carlat et de Murat.

The above mentioned city of Murat Cantal claims to be an ancient oppidum first mentioned under that name in 270:

Murat est l'une des villes du département dont l'origine se perd dans la nuit des temps. Son nom celtique signifierait " roc escarpé ", il apparaît pour la première fois dans l'histoire en l'an 270 lors de l'évangélisation de l'Auvergne. La ville est une place fortifiée, sa vicomté est déjà puissante et son château existe longtemps avant l'an 1000.

The two castles bearing that name:

This castle in now the commune of Saint-Étienne-de-Maurs in the Cantal. First mentioned as as Murat l'Arispe, then Murat la Rabe and Murat la Guiolein after 1258.
–– Château de Murat and Un peu d'histoire: Le château de Murat, Château de Murat

Fascinatingly, the castle Murat la Rabe was also called Murat-l'Arabe – literally "Murat the Arab". (See link above and confer to Jean-Baptiste Bouillet, 1799–1878. Dictionnaire héraldique de l'Auvergne. 1974 (eg p245/262, associating that naming to 1306) and one Guy de Murat apparently also had the nickname of 'the Arab'. But that seems to related not to his ancestry but to crusading adventures? Folk etymology now says it would derive from "Arabs once owning it", but that seems not to be true.

An actual French hydronym would then be: Murat (Aumance) (sadly not even on the French Wikipedia! But on maps and in a catalog.)

The name for small town of Labastide-Murat is an exception in the list above, as it actually was named after the general. It is also the home town of Joachim's ancestor Pierre Murat (1634).

This is however not the case for others on the list — and in the general case of a river being renamed historically speaking "very highly unlikely".

That gives quite ample opportunity for a "true frenchman" origin of a toponymic surname.

Wiktionnaire:
Etymology
Occitan Murat ("walled place, surrounded by walls")

Also compare the Latin murus with:

mur From Old French mur, from Latin mūrus, from Old Latin *moerus, *moiros, from Proto-Indo-European *mey- (“to strengthen”).

Apparently a family name of nobility already existed earlier:

Early Origins of the Murat family

The surname Murat was first found in Gascony (French: Gascogne), an area of southwest France bordering Spain, that was part of the "Province of Guyenne and Gascony" prior to the French Revolution, where this eminent family has held a family seat since ancient times.

Important Dates for the Murat family, covering the years 1261, 1455, 1500 …

Spelling variations of this family name include: Murat, Muratt, Murrat, Murratt, Muratte, Muratet, Murratet, Muret, Murret, Muron, Murron, Mur, Mureau, Murreau, Mureaux, Murreaux, Muraut, Murault, Murrault, Mureux, Lemur, Lamure and many more.

_{–– Murat History, Family Crest & Coats of Arms (Commercial, paywalled, thus uncertain trustworthiness.)}

One of those French nobles would then be Géraud Ier, vicomte de Murat ca 940 de Murat.

Conclusion

The French family name derives from the French place name.

Small addendum:

his very German first name…

I would say that first Joachim is not really a German name, although quite common there.

Joachim is derived from Hebrew and in French we see:

Étymologie
(ca. 1100) Du latin Joachim; de l’hébreu ancien יהויקים, Yehoyakim.
Prénom
Joachim \ʒɔ.a.ʃɛ̃\ , \ʒɔ.a.kɛ̃\ ou \ʒɔ.a.kim\ masculin

Prénom masculin. Joachim du Bellay (ca. 1522)