"If your dog is fat, you're not getting enough…

“A fat dog is likely to have a fat master” was cited in print in 1958. Dogs and their owners often eat at the same time, and they both often walk (exercise) together. The saying “If your dog is fat, you’re not getting enough exercise” has been cited in print since at least 2002 and is of unknown authorship. The saying has been printed on many gift items, such as T-shirts and posters.

27 January 1958, Evening World-Herald (Omaha, NE), “Business of Dog Insurance Goes to—Well, the Dogs,” pg. 3, col. 4:
Kansas City, Mo. (UP)—A fat dog is likely to have a fat master.
(...)
“If you’re fat yourself and your dog is fat,” Mr. Reabe says, “the chances are that your dog has caught your psychosomatic desire for too many calories.”
(Cordon Reabe, first canine actuary in the United States—ed.)

29 October 1962, Evening World-Herald (Omaha, NE), pg. 5, col. 3:
Fat or Nervous?
Take Look at Dog
MIAMI BEACH, Fla. (AP)—If your dog is fat or nervous, take a good look at yourself.

Google News
22 June 1979, The Free Lance-Star (Fredericksburg, VA), pg. 14, col. 3:
If dog is fat,
look at owner
CHICAGO (AP)—Fat Fidos tend to belong to fat folks and their health, like that of their owners, is endangered, says a director of the AAmerican Animal Hospital Association.

18 February 2002, Philadelphia (PA) Inquirer, “With four legs or two, you benefit from a walk,” pg. D1:
“If your dog is fat,” the saying goes, “you aren’t getting enough exercise.”

Baltimore (MD) Chronicle & Sentinel
This story was published on July 3, 2002.
Quotes About Dogs…
(...)
“If your dog is fat, you aren’t getting enough exercise.” — Unknown

Google Books
A Bark in the Park:
The 40 best places to hike with your dog in the Reno/Lake Tahoe Region
By Sherril Steele-Carlin
Montchanin, DE: Cruden Bay
2003
Pg. 28:
“If your dog is fat,” the old saying goes, “you aren’t getting enough exercise.” But walking the dog need not be just about a little exercise. Here are 10 cool things you can see around Reno and Lake Tahoe while out walking the dog.

Google Books
The Canine Hiker’s Bible
By Doug Gelbert
Montchanin, DE: Cruden Bay Books
2004
Pg. 109:
“If your dog is fat, yon aren’t getting enough exercise.”—anonymous

17 April 2005, San Antonio (TX) Express-News, “Exercise your pets for both your sakes ” by Cathy Rosenthal, S.A. Life:
A few months ago, I saw a sign on a veterinarian’s marquee that said: ‘‘If your dog is fat, you’re not getting enough exercise.’‘

Google Books
Secrets of a Top-Scoring Student:
What you can do to achieve great grades!
By Linus Tham
Singapore: Armour Publishing
2009
Pg. 118:
If your dog is fat, you’re not getting enough exercise.
~Author Unknown