I have an earlier answer on reactions to the Reformation in the (western) Islamic world, if you're interested!

• What was the Muslim world's view of the Protestant Reformation?

As a major power player in Europe, the Ottomans absolutely had a vested interest in the conflict between Catholics and Protestants--even if not strictly a theological one. They also had to contend with whether Catholics and Protestants in their eastern European territories qualified as dhimmis when they clearly hated each other and considered themselves to be different religions.

In spring of 1529, a group of gospel-minded German princes is drawing up the document of protest that will ultimately yield their ideology the label Protestant. At the same time, an Ottoman army is preparing to march on Vienna. Their mutual target? The Hapsburg Holy Roman Empire.

While Ottoman and Western navies fought for control of various Mediterranean islands, coastlines and shipping routes (the famous Battle of Lepanto is 1571), the Hungarian frontier was the front line for Ottoman ambition in Europe. Due to the decentralized nature of the HRE, that meant the Habsburgs headquartered in Vienna, rather than other German princes, were their primary target. At the dawn of the Reformation, the Habsburgs under Charles V pretty much directly controlled Austria, Spain and some of the Netherlands. This political situation shapes the Ottoman reaction to the Reformation.

To be clear, the German Protestants made no attempt to endear themselves to the Ottomans. In 1528, Luther himself published a treatise "On the War Against the Turks," in which he gave up on the idea of converting Muslims to Christianity and focused instead on defending Western Europe from the Turks. So it wasn't to Lutheran princes in Germany for religious reasons, but rather to Catholic France to whom the Ottomans reached out first for political reasons: France was the sworn Habsburg enemy.

This decision benefited everyone except Charles. As rumors of a true Franco-Ottoman alliance (which may or may not have resulted from their diplomatic efforts) helped distract some Hapsburg attention, Lutheranism in German gained a lasting foothold. As a result of that, when Protestant princes took up arms against Catholic in the Schmalkaldic Wars, the Ottomans got to take a necessary breather. Since the Protestants actually lost this set of wars, it turned out that the (Catholic) Empire's distraction by the Ottomans ended up cracking open the door for Protestantism's long-term survival: a Lutheran force was able to score a minor victory that forced the Peace of Augsburg, the treaty that mandated each HRE principality would follow either Lutheranism or Catholicism as dictated by its prince.

Around this time, the Ottomans made diplomatic inroads with other enemies of the Habsburgs, namely Protestants in Elizabethan England and Calvinists in the Netherlands. Letter exchanges reveal a diplomatic emphasis on religious mutual ground, especially a common rejection of "idols" (Catholic veneration of images). This is diplomacy, not a real effort at interreligious dialogue. Ottoman sultan Murad III expressed his belief that the Protestant rejection of images was a step on the right path:

As you, for your part, do not worship idols, you have banished the idols and portraits and ‘bells’ from churches, and declared your faith by stating that God Almighty is One and Holy Jesus is His Prophet and Servant, and now, with heart and soul, are seeking and desirous of the true faith.

Queen Elizabeth, for her part, emphasized the Protestant/Muslim mutual worship of

that God (who onely is above all things, and all men, and is a most severe revenger of all idolatrie, and is jelous of his honour against the false gods of the nations)

so you can see both parties neatly ignoring rather than working to reconcile the Jesus Problem in order to establish a basis for international relations. The diplomatic efforts opened the door to a very cool letter exchange between Elizabeth and the Ottoman valide sultan (queen mother!) Safiye, but more importantly in the scheme of things, to an Ottoman-Dutch alliance that aided both parties tremendously.

Initially, of course, Ottoman promises of aid to Dutch Protestants against their Spanish Catholic overlords were mostly lip service, and a few waves of the Dutch Revolt failed. More important than direct military aid from the Turks to the Dutch, however, was--again--the distraction they provided for their enemies. The ongoing skirmishes between Spain and the Ottomans diverted Iberian resources away from the Low Countries, and eventually allowed the 1570s wave of rebellion to stick its landing. For their part, the Ottomans gained crucial diplomatic and trade ties with a rising world power.

Transylvanian Hungary hosted the only sizable Western Christian population under Muslim control. Here, the Ottomans extended their recognition of (Catholic) Christians as dhimmis or people of the book to Protestants as well. As the Reformation in Hungary tinged Calvinist, the Ottomans proved great supporters of that church in their territory, again seeing it as a weapon against Habsburg Vienna. They were also happy to continue extending the same status to Catholics. Refugees of inter-Christian conflict as well as problems within a branch of Christianity or even a specific Catholic order all took shelter in Ottoman territory. And under the crescent, which permitted religious practices but not religious construction by non-Muslims, diverse Christian populations across the OE ended up rotating holy services in the same church buildings.

So overall, this is a political story much more than a religious one. The diplomatic ties tentatively established through initial epistolary diplomacy and eventual ambassadorships did not prove completely reliable under duress. Nevertheless, Western and Eastern parties alike--Protestant, Muslim, and sometimes even Catholic--found it useful to keep the possibility of alliance alive. God save the king (and his useful army), they could all agree, as long as that king was not also Emperor.