The big question for Zubin Mehta in 1978 was whether he had the star quality to conquer New York.

There is now a sign in his office at Lincoln Center suggesting that he has taken New York in three years.

It is an enlarged and framed New Yorker cartoon. A genteel-looking couple is having martinis and the woman is saying to the man:

"I had the strangest dream last night. Muskie became CBS News anchorman, Cronkite became secretary of state, Mondale took over from John Chancellor, Dan Rather became attorney general and Ronald Reagan replaced Zubin Mehta as conductor of the New York Philharmonic."

There was no question of the dark and handsome Mehta's physical star quality. The issue was leadership of the New York Philharmonic, following in the footsteps of Mahler, Toscanini and Bernstein, just to name a few.

In his years directing the Montreal and Los Angeles orchestras, it was his brooding presence -- particularly on television and film -- that most quickly caught the popular audience, particularly the women in that audience. CBS Records even has marketed a recent recording of Beethoven's "Eroica" with your own "special limited edition, full-color 22" x 22" Zubin Mehta poster," an almost life-size version of the Indian conductor's dark eyes, arched brows and flowing locks. Asked about the poster, Mehta says, as if a bit embarrassed, "Oh, you know, that thing is really nothing." But to CBS, whose Philharmonic sales have risen sharply with Mehta, it's hardly nothing.

Mehta, who conducts the Philharmonic at Wolf Trap tonight in the second of two performances here, has been rising in the musical world for 20 years. He is now 45, and time is running out for some of the wizened critics to end their suspended judgments about his potential, saying typically, "Well, he's glamorous, and of course he's brilliant. But, then, he's got so much growing up to do."

Certainly, Mehta is not in a position to deny that he's upset a lot of people along the way. In late 1967 he took umbrage at the New York establishment with this blast:

"The musicians of New York step all over conductors. A lot of us think, why not send our worst enemy to the New York Philharmonic and finish him off once and for all."

Normally that's called burning your bridges. But three years ago Mehta left the Los Angeles Philharmonic, after 16 years, to join the New York orchestra as its new music director.

This Monday he was rehearsing the players at New York's Tully Hall and the mood seemed almost collegial. Mehta was meticulously preparing, in small segments, the last movement of Brahms' Third Symphony, one of the works to be featured on a three-week national tour.

Physically he is a little pudgier than before, and utterly assured. The august members of the orchestra he once lashed out at address him as "Zubin," just as they speak to Bernstein as "Lenny."

It has been six years since the Philharmonic last played the Brahms and Mehta says that there has been a 50-percent player turnover during that period. He is particularly concerned with violin bowings. He tells the orchestra, "The thing is to start with the tip of the bow so that at the right moment you can have the lovely sound that comes from the second half of the bow."

At one moment in the rehearsal his mischief takes hold. During a brief pause, a horn emits a sound that might best be described as unmentionable. Mehta barks, "What is that?" The player cracks, "That had nothing to do with me." And everybody laughs.

Later, in his office at Avery Fisher Hall, Mehta plops on a sofa and props his legs on the coffee table. Beside him are pictures of his children, his wife (his second, actress Nancy Kovack), and one of the indisputable musical giants of the century that is inscribed: "To Zubin, this favorite son of the gods, with my friendship, love and admiration. Arthur Rubinstein."

On a wall is a framed clipping from page 1 of the April 29, 1936, New York Times reporting the retirement of Arturo Toscanini from the directorship of the philharmonic. "That was the day I was born," he declares. "Can you believe it? Someone had a copy of this thing and they sent it to me."

If Mehta's birth date is an interesting coincidence, his birthplace -- Bombay -- is genuinely remarkable for a major conductor.

His father, who now lives in Los Angeles, was a musician in a country and city where classical music has never been as important as in the West. The Mehtas are Parsis, a small elite originally of Persian origin and Zoroastrian in faith. Bombay, by no means a musical center, was Mehta's home until he went to study music in Vienna at 18 ("I had a cousin there"). Mehta thinks his father made a conducting career possible.

"He made himself the original self-taught Indian musician. He had no background, and how he came to music God knows. He was given a violin and he showed talent and he used it to play the Beethoven violin concerto. I tell you, he was a miracle in a cultural sense. He founded an orchestra, he was the concertmaster and he formed a quartet."

One upshot of this unlikely background for the younger Mehta was that his arrival in Vienna to study brought about "such a case of culture shock that I'm just not verbal enough to describe it. I already knew a lot of music, but it was by four hands on the piano and records, everything. And I feel that one of the advantages was that I heard the Brahms First played by an orchestra there, for the first time in my life, conducted by Karl Bo hm with the Vienna Philharmonic right there in the Muzikvereinsaal.

"I just ask you to imagine what that was like after growing up in Bombay with an absolutely fourth-rate orchestra that had never played anything but, you know, little ditties, never played a Brahms symphony, and I come to Vienna, I hear one of the great orchestras of the world, one of the great Brahms conductors and in literally the best hall in the world.

"I knew the Brahms already, but I didn't know it like a professional. You understand, there is a difference. It wasn't like I had heard it first done poorly in some little European town . . . and had to relearn it. I had never heard it at all. I feel like I heard it the right way from the beginning. And it was the same with Mozart operas with Bo hm."

One charge a few critics have leveled at Mehta is that he tried to go too far too fast. "I really don't rush; I just appear to," he protests.

Then he recalls an exception. "I did the Beethoven Ninth before I did the 'Eroica.' I can't tell you why, but sometimes I am in awe of a work. In the case of the 'Eroica,' that was absolutely the point. The Ninth I had studied in school. I had played it in the orchestra. I had listened to umpteen great conductors' rehearsals. I read about it. I did everything I could to come close to it. And yet the first time I conducted it, I tell you it was really terrible. You stand in front of an orchestra and you want to expound, and nothing comes out of you and you stand there like an ignoramus. I didn't go back to that orchestra the Czech Philharmonic for many years. I was so ashamed of myself."

That experience taught Mehta to be more cautious. He has canceled a "Die Meistersinger" at London's Royal Opera in 1982, "and they are mad, but the only time I could work on it was during my two weeks that just finished in Los Angeles. And I didn't want to. Anyway, I can't prepare 'Meistersinger' in two weeks."

Another work that Mehta hasn't scheduled yet is Bach's B-Minor Mass. "I sort of have to know things and they have to wallow and sort of circulate in my system somehow. I've done the St. Matthew Passion often. But it's different; the St. Matthew Passion is a story. It's a lovely narration. The B-Minor Mass is a geometrical, trigonometrical puzzle. And I'm talking from a musician's standpoint, I'm not talking about the spiritual content of the work. But," he adds, raising his voice, "until one digests both categories, both aspects, one shouldn't perform it."

The New York critics have been generally friendly toward Mehta. That upstart "Zubie-Baby" label that so rankled him in his Los Angeles days (he maintains the phrase, which he calls "atrocious," actually is the invention of an Eastern critic) is less often heard. His original three-year Philharmonic contract has been extended through the 1985-1986 season.

Mehta grows impatient at times with the public's and the media's occasional inclination to stereotype dashing figures like him, Leonard Bernstein and Herbert von Karajan as stars first and musicians second -- meanwhile leaving aside great but grayer men such as Bo hm. "It would be wonderful some day to put our recordings side by side with the undashing type and ask people to guess which is which. It's a visual thing and Lenny has really had to take a lot of this over the years. Lenny also had to do with Broadway and with the Hollywood movies, so he had more of a problem. That Broadway-Lenny notion was terrible and unfair. A more serious a musician than him you can't find. Just listen to his new recording of Beethoven's Missa Solemnis."

Mehta and Bernstein also share a proprietary involvement with the Israel Philharmonic, of which Mehta has been music director since 1969.

"I pulled a bad one there on election night recently when I announced that Peres, not Begin, had won. There was jubilation in the concert hall, though, because the people who come to concerts voted for Peres. The others voted for Begin."

Mehta adamantly maintains that first and foremost he is, and always will be, Indian. "Those roots you just can't take away. We have a very strong family tradition. My food is Indian. My language is Indian. My passport is Indian. People here should know more about India. You know, there really are no snakes on the streets.

"You would have been convinced about how I feel if you had seen me Sunday walking in a parade on Madison Avenue to celebrate the Indian Fourth of July," he declares. "The 15th of August is our Independence Day from England, and you know I was taken to the reviewing stand with Mayor Koch. And there were about 30,000 Indians there. I was so at home and I was asked to say a few words. I screamed, 'Do you remember what it was like that day?' That day was such an awakening. And maybe the English left too soon. They should have stayed a few more years. It was too soon after the war and we didn't have enough people to govern the country. Now I say this. But in those days it was a great relief, an explosion."

But if Mehta sees himself personally as primarily an Indian, as a conductor he sees himself more as an American. The Mehtas have kept their palatial home in Brentwood, and go there for rest and recreation. "I love California, I never plan to give the home up," he says. "We have a town house here in New York. But that's just not the same.

"Out there I can do so many things. That's why I won't be conducting the Los Angeles Philharmonic for a few years. I can be freer. Why, last week I was at a baseball game, in the owner's box with Cary Grant, and you know what we talked about? Cricket."