This has happened only once in the modern era (1900+) of major league baseball, a pitcher throwing a no hitter in his first appearance for a team.
The man to accomplish it was the Cubs Don Cardwell, on May 15, 1960.
Two days earlier, the Cubs had obtained Cardwell, along with my pal Ed Bouchee, in exchange for Cal Neeman and Tony Taylor. As a Phillie, he had an unspectacular 17 - 26 record over 3 1/2 seasons. I guess the Cubs figured he had potential and so they made the deal.
Cardwell's first start was the second game of a double header against the Cardinals at Wrigley Field. In the first inning, he retired the leadoff man, then walked the second batter, Alex Grammas. That would be the only runner for the Red Birds. Cardwell retired the next 26 in a row, striking out seven along the way.
The final out was dramatic, as left fielder Moose Moryn made a diving catch of a sinking liner off the bat of Joe Cunningham. You can hear the catch as called by Cubs radio announcer Jack Quinlan here (it begins at the 24 second mark).
The back of Cardwell's 1961 card mentions the no-hitter,

Topps 1960 card of Moose Moryn. I wonder who had the idea to use that pretty shade of pink? Do any of this year's Heritage cards have the same shade?

Unfortunately, the no hitter didn't propel Cardwell onto greatness. He went 7-14 the rest of the year with the Cubs. His best season was 1961, when he went 15-14 for the Cubs. He slumped in 1962 to 7-16.
Traded after the 1962 season, he bounced around with the Pirates, Mets, and Braves. His career won loss record was 102 - 138. But on an exciting day in May, 1960, he did something no other major league pitcher can claim. And that has to be worth something.
Here are a couple more cards to add to the 1969 Update set.
Back up catcher Ken Rudolph

One of the many centerfielders, Oscar Gamble
In this post I talked about creating cards for the 40th anniversary of the Cubs 1969 flop. The choices were to make 2009 styled cards or update the original 1969 cards with action shots like today's cards. Thank you to all who left comments. The consensus was to do both. And so I will.
Today, I've got the 1969 cards ready. There are 12 of them, featuring the main players of the team. I've got seven of the eight regulars (no centerfielder -since the Cubs had no decent centerfielder that season; Adolfo Phillips, Don Young, Jim Qualls), two starting pitchers, the closer, the utility infielder, and the manager. I was able to find action shots of most of the players. And most of the pictures are from the 1969 season. It is fairly easy to identify pictures from that year because teams wore the Baseball Centennial patch that season only.
Without further ado, I present the updated 1969 Topps Chicago Cubs:
First baseman, Mr. Cub Ernie Banks, taking a swing at Wrigley. Check his right sleeve for the centennial patch.

Second baseman Glenn Beckert

Shortstop Don Kessinger

Third Baseman Ron Santo

Left Fielder Billy Williams

Right Fielder Jim Hickman

Catcher Randy Hundley

Pitcher Fergie Jenkins

Pitcher Ken Holtzman

Relief Pitcher Phil Regan

Utility man Paul Popovich. This card is my tribute to the airbrushing of the era. The best picture I could find of Popovich was with him in a Dodger uniform, although the shot is at Wrigley Field. So I airbrushed the "LA" off of his hat, just like Topps would have.

And the last card in the set, Manager Leo Durocher.

Three Hall of Fame players (plus Santo, borderline), a Hall of Fame manager, and a lousy second place finish. Damn Mets!