QAnon Cheers Republican Attacks on Jackson. Democrats See a Signal.

The online world of adherents to the QAnon conspiracy theory sprang into action almost as soon as Senator Josh Hawley tweeted his alarm: that Judge Ketanji Brown Jackson, the Biden administration’s Supreme Court nominee, had handed down sentences below the minimum recommended in federal guidelines for possessing images of child sexual abuse.

“An apologist for child molesters,” the QAnon supporter Zak Paine declared in a video the next day, on March 17, asserting without evidence that Democrats were repeatedly “elevating pedophiles and people who can change the laws surrounding punishment” for pedophiles.

By Wednesday, as Judge Jackson appeared for the third day before the Senate Judiciary Committee, claims that she was lenient toward people charged with possessing the illegal imagery had emerged as a recurring theme in her questioning by Republicans.

Never mind that those sentences did not come up at Judge Jackson’s confirmation hearing last year to a federal appeals court, that other judicial nominees have faced no questions about similar sentencing decisions, or that a former federal prosecutor called the allegations “meritless to the point of demagoguery” in the conservative National Review.

The line of attack has set off a new debate over the Republican Party’s stance toward QAnon. A White House spokesman this week accused Mr. Hawley of pandering to the conspiracy theory’s believers among his party’s rank and file, calling his comments an “embarrassing QAnon-signaling smear.” Conservatives, in return, blasted the Biden administration for invoking the specter of QAnon for its own political agenda, to fire up the Democratic base without addressing the questions.

“Left Invokes QAnon After Josh Hawley Exposes Ketanji Brown Jackson’s Soft Record on Child Sex Offenders,” declared a headline on the right-wing website Breitbart that was widely shared this week in QAnon circles.

A spokesman for Senator Hawley declined to comment on his motivations.

Although few QAnon followers appeared to take notice of Judge Jackson’s sentencing record before Senator Hawley’s tweets, her judicial career had touched the roots of the conspiracy theory: an earlier internet myth known as Pizzagate.

That debunked theory held that Satan-worshiping Democrats were trafficking children out of the basement of a Washington restaurant, and in 2017 a believer armed with an assault rifle stormed in and fired his weapon. Judge Jackson, as a district court judge, sentenced him to four years in prison, saying his actions “left psychological wreckage.”

The QAnon conspiracy theory was born a few months later when an anonymous writer — often signing as Q — elaborated on the discredited myth that a cabal of top Democrats was abusing children. Q purported to be a top official close to President Donald J. Trump and asserted that the president was waging a secret war against the cabal.

Slogans about protecting the children became catchphrases that QAnon adherents used to identify one another, and their bizarre fantasy — initially encouraged by far-right news outlets, then promoted by a ring of social media influencers — appeared to spread widely among Trump supporters. At least two Republican lawmakers elected in 2020 had made statements supportive of QAnon, and prosecutors say that many people involved in the Jan. 6, 2021, attack on the Capitol subscribed to the theory.

Among those now echoing the Republican allegations about the judicial nominee, in fact, is Ron Watkins, a former website administrator who is widely believed to have played a major role in writing the anonymous Q posts. Mr. Watkins, who has denied any part in the Q messages, is running for the Republican nomination to an Arizona congressional seat, largely on the strength of his QAnon association; this week, he qualified for the ballot.

“Judge Jackson is a pedophile-enabler,” Mr. Watkins wrote Wednesday on social media. “Any senator who votes to confirm her nomination is also a pedophile-enabler.”

QAnon Telegram channels on Wednesday grew increasingly agitated. “She has committed unbelievable crimes against humanity with her judgeship,” one user wrote. “If she gets confirmed the victims remain victims & trapped in the misery bestowed on them,” said another. Some talked of violence.

Polls suggest that QAnon supporters have continued to make up a significant portion of the Republican base even after Mr. Trump’s departure from office contradicted Q’s predictions. One poll last October found that about 60 percent of Trump voters had heard of QAnon, and 3 out of 10 of those Republicans viewed it favorably.

Yet the same poll found that Democrats were far more likely to say they had heard a lot about QAnon and also overwhelmingly to reject it, and other polls, taken after the attack on the Capitol, indicated far more widespread condemnation. Democrats thus have much to gain politically from linking the name “QAnon” to Republicans questioning a Supreme Court nominee, the polls suggest, but individual Republicans might benefit by signaling to QAnon supporters without explicitly naming the movement.

“You wouldn’t talk about the extreme stuff, but you would talk about how people in elite power are enabling traffickers,” said Bond Benton, an associate professor at Montclair State University who has studied QAnon. “That is a secret handshake to the Q crowd.”

Other conservative commentators have noted that soft-on-crime or soft-on-sex-crime accusations against politicians or judges have long resonated widely with voters regardless of connection to QAnon, disputing the accusation that the Republican questions are any kind of covert signal.

Others on the right have also accused Democrats of employing their own dog whistles — notably when Amy Coney Barrett, a practicing Catholic and now a Supreme Court justice, was nominated to an appeals court. Many conservatives have said that they heard a covert appeal to anti-Catholic or anti-religious bigotry when Senator Dianne Feinstein, a California Democrat, told the judge that “the dogma lives loudly within you.”

Jim Manley, a former top aide to the Senate Democratic leadership who helped wage a half-dozen battles over Supreme Court confirmations, said that party elders often understand the Senate math makes confirmation highly likely and prefer to get it over quickly, without mudslinging that could alienate moderate voters — in this case, by evoking QAnon.

“But I learned the hard way that there are always some in the caucus — especially those who may be thinking about running for president — who are going to want to throw some red meat to the base,” Mr. Manley said. “They just can’t help themselves.”