Last month, Germany began deploying an army battle group to Lithuania, the first of the North Atlantic Treaty Organization troops to arrive to bolster the defenses on the alliance’s eastern border with Russia.

It isn’t an overwhelming display of force. The initial German contingent is 460 troops, supplemented by a few hundred soldiers from Belgium, the Netherlands, Luxembourg and Norway.

Some current and former American officials derided the unit as a “Frankenbattalion,” calling it an example of Germany’s failure to shoulder its fair share of the NATO burden. While Germany defended its plan and the U.S. has dropped its official complaints, it illustrates the tensions coursing through the alliance as the Trump administration prepares to push Europe for more defense spending.

“We have been complaining since 1949 that European allies aren’t doing enough,” said Jim Townsend, who served in the Pentagon during the Obama administration. “But for Germany it has been particularly problematic in the last 10 years.”

NATO is at a crossroads. Having helped keep the peace in Europe for more than 70 years, the 28-nation alliance is being sharply challenged by Russian aggression in Ukraine, and by President Donald Trump, who has called the organization obsolete and argued it should focus on counterterrorism.

This week, top officials from the new U.S. administration come to Europe for a NATO meeting in Brussels and a security conference in Munich where questions of the group’s missions and finances will be on sharp display. How member countries resolve their differences will go a long way toward determining NATO’s future and usefulness.

Mr. Trump has signaled he will put new muscle behind America’s long-standing demand that Europe spend more on defense. “We only ask that all of the NATO members make their full and proper financial contributions to the NATO Alliance, which many of them have not been doing,” Mr. Trump said in a speech last week at MacDill Air Force Base in Tampa.

On Tuesday, NATO Secretary-General Jens Stoltenberg announced that NATO allies in Europe and Canada raised their defense spending by $10 billion last year, a 3.8% increase that is bigger than allied officials initially expected.

The U.S. spends $664 billion annually on its military, or 3.61% of GDP, tops in both categories of any NATO country. Spending by other NATO members ranges from nothing, by Iceland, to $60.3 billion by the U.K.

Germany, the economic powerhouse of Europe, spends around $40 billion, or 1.2% of its gross domestic product, on defense. The U.S. has been pushing for Germany to hit 2% of its GDP on defense spending for more than a decade.

More recently, Chancellor Angela Merkel has been trying to push her pacifist-minded country to close the gap. In November, the German Parliament made the biggest increase in military spending in more than a decade, raising the defense budget 8% to €37 billion ($39 billion), and German government officials say they will push for more increases in the coming years. Alliance officials note if Germany was to meet NATO’s 2% of GDP goal, it would require tens of billions of dollars in extra spending each year.

Administration officials, including Vice President Mike Pence and Defense Secretary Jim Mattis, will come to Europe this week to deliver a message of reassurance, but also to note that defense spending must rise.

German Defense Minister Ursula von der Leyen—perhaps the German politician most supportive of increased military spending—visited Washington on Friday to detail her country’s defense work and called demands for a larger defense budget appropriate.

“It’s a fair demand,” she said during her trip to Washington. “If we want to jointly master the crises in the world, namely the fight against terrorism, and also put the alliance on solid footing, then everyone has to pay their share.”

Some German policy makers say the U.S. appears to be overly focused on the 2% number, given Germany’s engagement on the ground from Afghanistan and Iraq to Mali and Kosovo.

U.S. officials said they aren’t worried about spending for spending’s sake but see a real military gap—in personnel and materiel.

Core to Europe’s defense is the U.S. contribution. The U.S. has 35,000 military personnel in Europe, mostly in Germany, including two Army infantry brigades. The U.S. has bolstered its force with a heavy tank brigade with 3,500 troops and 87 tanks. It also maintains tanks and artillery at sites around western Europe.

That said, NATO plans for the defense of Europe rely heavily on Germany, which will be required to have six heavy infantry brigades ready to reinforce Poland or the Baltic states in the event of a conflict with Russia, according to Western officials briefed on requirements. NATO is pressing for more tanks, long-range artillery, ground-based air defense systems, aerial refueling planes and other equipment from Germany, officials said.

Some German lawmakers have questioned if the alliance is mistakenly preparing for yesterday’s battles.

“I do believe that we need to do more in terms of equipment for the army,” said Rainer Arnold, the top defense-policy expert in parliament for the center-left Social Democrats. “But we must be careful about believing that Europe will be defended in a great tank battle—that doesn’t conform with today’s military technology.”

Mr. Arnold said the 2% goal is “a utopia,” because Germany would struggle to spend such a large sum, and many of its neighbors would react warily if Berlin dramatically increased defense spending.

NATO officials say the new requirements are a necessary calibration to respond to the threat from Russia. Alliance officials say they are making investments in drones and cyberdefenses, but the Russian military buildup must be immediately countered with the kind of heavy military equipment that will ensure Moscow realizes any incursion would ultimately fail.

To a certain extent, all of NATO’s European allies have been caught by surprise as the threat has moved from counterinsurgency and low-intensity combat to preparing to defend against Russia. For years NATO urged European allies to tailor their forces to the kind of peacekeeping missions they were conducting in Kosovo or the kind of fighting and training missions they conducted in Afghanistan.

Germany had more than 2,000 Leopard 2 battle tanks in its arsenal during the Cold War and 800,000 military and civilian personnel in its armed forces after the Berlin Wall came down. Successive rounds of cuts whittled the military down to 177,000 service members, a maximum of 56,000 civilians, and a goal of just 225 Leopard 2 tanks.

NATO pushed European countries to model their forces on the British, de-emphasizing tanks and focusing on light deployable forces. Now the alliance has shifted gears demanding the country once more build up its heavy forces. Germany has already started to follow suit, moving last year to add thousands of new military positions and refurbish scores of decommissioned tanks.

The U.S. military’s focus on Germany as the potential game-changer  is in part because of the size of its economy, but it is also because of the quality of its equipment.

U.S. military officers often speak with unhidden jealousy about German Panzerhaubitze 2000, viewed as perhaps the most advanced artillery system of its kind.

The problem: Germany has only about 90 of the artillery in service, having sold off 16 to Croatia and 21 to Lithuania. According to U.S. officials, German troops must share large artillery pieces for training, because they don’t have enough to go around.

German officials said artillery pieces are sometimes unavailable because of maintenance needs and that the military is now expanding its stock, with 12 new Panzerhaubitze 2000s slated to be delivered this year.

The U.S., by contrast has 5,923 artillery pieces, including 969 of its most advanced system, according to the International Institute for Strategic Studies. This year the U.S. will be moving a part of that arsenal, a brigade worth of artillery, to storage bunkers in Europe.

The small number of troops deployed to Lithuania raised questions among some U.S. officials about whether the German military is large enough.

Lithuanian officials said they aren’t worried and pointed to the swiftness of the German deployment, which arrived ahead of the British and Canadian forces headed to Estonia and Latvia.

German officials argue a multinational battalion is a better deterrent: Russia would know it would cross not one ally but many if it made a move in Lithuania. Allied ambassadors backed the German plans, and the U.S. stopped pushing the issue, saying it was settled.

Officials are hoping for a change in Germany’s pacific public attitude about military action, shaped by the shadow of World War II. Polls show growing concern about Germany’s security in the wake of the Ukraine crisis and high-profile terror attacks. The German military’s Center for Military History and Social Science found half of Germans in a poll last year wanted the defense budget to be increased, compared with just 19% in 2013.

In November, the German military premiered a $7 million, 82-episode reality show on YouTube called “The Recruits,” which follows the exploits of a group of 12 trainees as they navigate basic training on the Parow Naval base on the Baltic Sea.

The slickly produced show doesn’t obscure the challenge for Germany. In one episode, Petty Officer Second Class Carl Scholwin, who has spent 10 years in the military, laments that service members in his country “don’t get the recognition that they really should get.”

Asked later by The Journal about what it is like to return from a mission abroad, he said: “In Germany, you get left by the roadside. You’re not really noticed.”