A drug popularized under the trade name of Quaalude has become a public health menace on a scale with heroin, according to Drug Enforcement Administration officials.

DEA surveys of morgues and emergency rooms show that the drug, methaqualone, is the most-used illegal drug next to marijuana and causes more injuries and trauma than even heroin and cocaine in 13 major cities throughout the country.

But, unlike marijuana and heroin, the sources of the drug methaqualone are legitimate drug and chemical companies.

There is only a small legitimate medical use for the drug, which is prescribed by doctors for use as a sleeping pill, but it is manufactured in huge quantities by legitimate companies in Europe, according to Gene Haislip, director of compliance and regulatory affairs for the DEA.

More than half, or possibly much more, of the legal production goes to illegal abuse, Haislip said. This kind of legal overproduction, followed by illegal diversion, is becoming an increasing problem for many drugs, from stimulants to sedatives, he said.

To combat a spectacular rise in the last three years in the use of methaqualone, which is abused mostly by teen-agers, the DEA has shut down for four years the one source of the raw chemical in the United States, Haislip said.

He said the DEA also has obtained an agreement from the Hungarian government to close down that nation's entire production, estimated at 32,000 pounds a year.

Acording to a report published yesterday in the Journal of the American Medical Association, adolescents on methaqualone are arriving at emergency rooms and morgues in record numbers, especially in Florida and Texas, where most of the drug comes into the country.

Methaqualone, like other sedatives, can kill if taken in too great a quantity or if taken with alcohol. A large proportion of the emergency room cases are those of teen-agers who have taken too much, mixed the drug with alcohol, or have run a car off the road under the influence of the drug.

In Broward County, Fla., a study was done among 356 drivers arrested for "driving under the influence" but hwo apparently hadn't had enough alcohol to account for their condition, according to the journal. Drugs other than alcohol were found in 295 of the drivers, and in 93 percent of the cases, it was methaqualone.

Fewer than 9,000 pounds of methaqualone are produced legally each year in this country. But DEA estimates that about 200,000 pounds or more make their way to the streets, a large percentage of that coming from legitimate chemical companies in Germany and Austria by way of South America and Mexico.

The DEA became alarmed about the pills, called "ludes" in street jargon, when their random drug seizures began turning up large amounts of the drug.

In 1978, the DEA seized 1,400 pounds of methaqualone; in 1979 it was 17,500 pounds and last year it was 27,692 pounds. The seizures for this year had already reached 71,500 pounds by May, but that total includes drugs seized under a new DEA program of planned raids.

Haislip said that the chief route of the drugs is from German chemical manufacturers to legitimate brokers in Hamburg, who sell the chemical to those who press it into pills and then smuggle them into this country.

Until the DEA brought administrative proceedings against the Ganes Chemical Co., the only U.S. manufacturer of the raw chemical, the company produced all the methaqualone allowed in the United States legally, about 8,800 pounds. It is made into tablets and distributed by Lemmon Co. of Sellersville, Pa., the only distributor of the drug in this country, and according to the company, the only legitimate distributor in the world.

A spokesman for the Lemmon Co., who asked that his name not be used, said he had no way of knowing how much of Lemmon's production is diverted to illegal use, although the DEA's Haislip says there is "some substantial diversion."

Lemmon's spokesman said, however, that he believes there is no other company in the world making methaqualone for legitimate medical purposes, which means that "all the methaqualone made by companies outside the United States is going to illegal use."