As my flair suggests, they definitely had a lot of problems with the Emperor being Catholic.

Unlike /u/Alex_Baratheon suggests, the Emperors themselves (who were all Catholic) were not mostly (never, I’d argue) the sole instigators of conflict, although Protestant histories—which still dominate the English-speaking popular discourse—have traditionally claimed this to be the case. The reality is pretty complicated, as religious issues were inextricable from political issues in pre-modern Europe. Note: I'll get to the specific question of the Protestant Electors once I lay out the basic story.

Following Luther's posting of the Ninety-Five Theses in 1517 and the formulation of new doctrines in the 1520s, Protestantism began to spread among the Empire’s princes and some of its population, particularly in central Germany. The Holy Roman Empire, famously, contained hundreds of effectively independent (meaning they answered only to the traditionally weak authority of the Emperor) entities, and the secular rulers found themselves drawn to Protestantism for a mixture of reasons. Some became Protestants out of sincere belief, of course, while others found the ability to absorb church land and revenues more appealing; for still others, Protestantism allowed princes to increase their authority by donning the mantle of religious authority. Most probably felt some combination of these attractions, though some resisted Catholicism completely, most notably the Bavarian branch of the Wittelsbach dynasty.

Charles V, who was Holy Roman Emperor from 1519 until his abdication in 1556, had made efforts to condemn Protestantism and contain its spread, beginning formally with the Edict of Worms in 1521. However, wars with France and the Ottomans kept his focus for over two decades after the Edict, which allowed Protestant princes to consolidate their control over their local churches, and organize militarily and politically to fight for what amounted to legal separation from the Empire. This organization was called the “Schmalkaldic League”, and had grown to contain the three Protestant Electors (Brandenburg, Electoral Saxony, and the Palatinate), as well as the powerful Landgrave of Hesse and numerous Imperial cities, by the time the first major armed conflict between Protestant princes and the Emperor broke out in 1546.

The Schmalkaldic War, as the brief conflict is known, resulted in an embarrassing loss for the Protestants. The Saxon electorship was passed to Duke Maurice, the Saxon Elector John Frederick I’s younger cousin. However, Charles V’s unwillingness to free Landgrave Philip I of Hesse and John Frederick even after their complete surrender angered the Protestant princes, and war began again in 1552. The ensuing “Princes’ War” saw France intervene militarily on the Protestant behalf for the first time, and French troops besieged the western border of the Empire while the Protestant princes forced Emperor Charles V out of Germany. Defeated, Charles V eventually agreed to the Peace of Augsburg (1555), which granted the Protestant princes the legal right to practice their new faith and effectively established the principle of Cuius regio, eius religio, which meant that secular princes could choose whether Catholicism or Lutheranism would be the legal religion in their territory. The Emperor abdicated the next year, splitting his Spanish and Austrian lands and thereby weakening the concentration of Habsburg authority. This may seem like a pretty satisfactory conclusion, but the religious question in Germany would remain unsettled for nearly another century, over thirty years of which would see religious warfare.

Now, at last, we get to the Electors. Because the Schmalkaldic League intended to subvert the Imperial process completely, the status of “Elector” did not matter to the early Protestant princes. However, with the Peace of Augsburg, Protestant princes now sought to influence Imperial law and policy to favor their cause. However, until the 1580s, it seemed that only three of the seven Electors would ever be Protestant: Saxony, Brandenburg, and the Palatinate. None of the other four seemed likely candidates to switch sides: Catholic bishops ruled Mainz, Trier, and Cologne, and the (elected) King of Bohemia was traditionally the Habsburg Emperor himself. The two times one of these four Catholic electors seemed likely to become Protestant—once in Cologne when its prince-bishop converted, and secondly when the rebelling, overwhelmingly Protestant nobility of Bohemia elected the Elector of the Palatinate to be its new king. The former inaugurated the Cologne War, as the Peace of Augsburg did not grant bishops the right to determine the faith of the territory under the policy termed “ecclesiastical reservation”. The Catholic side decisively won the Cologne War, and Cologne retained a Catholic bishop. This outcome, combined with aggressive re-Catholicization of Protestant nobility within the Habsburg territories, convinced many Protestant princes, particularly the now-Calvinist Electors of the Palatinate, that the Peace of Augsburg did not protect them sufficiently from the Catholic Habsburg threat. This led to the formation of the Protestant League in 1608, which was answered by the Catholic League in 1609.

Events came to a head in 1618, when the Emperor died and the Bohemian nobles (who elected their king) shortly thereafter launched a revolt. The Elector of the Palatinate, now a young Calvinist named Frederick V, lobbied heavily to stall the election of a new Emperor until the Bohemians had chosen a presumably Protestant King to serve as elector. This was the closest Protestants would come to electing a Protestant Emperor. However, Frederick V’s election as King-Elector of Bohemia, which would have theoretically made the vote 4 to 3 in favor of a Protestant Emperor (presumably Frederick himself) came too late to prevent the election of Archduke Ferdinand II of Austria, the Habsburg patriarch, as the new Emperor. Frederick’s decision to take the crown of Bohemia resulted in Ferdinand II removing him by force, and the war sparked by Frederick’s utter defeat waged on for thirty years (this is, of course, the Thirty Years War) until, after intervention by a list of foreign nations including France, Spain, England, Denmark and Sweden, the Peace of Westphalia finally laid religious conflict to rest in the Empire.

Main Sources:

Peter Wilson, Europe's Tragedy: The Thirty Years' War.

Thomas Brady, German Histories in the Age of Reformations, 1400-1650.