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The colloquial phrase to pee (also to wee, to piss) on a stick means:
– to take a pregnancy test of a type involving urinating on a disposable plastic stick which immediately indicates the result;
– also, more generally: to take any of various other diagnostic tests of this type—cf., below, quotation 3.

This phrase occurs, for example, in a review of Mom! A New Musical, produced at the Burnsville Performing Arts Center’s Black Box Theatre, in Burnsville, Minnesota—review by Liz Rolfsmeier, published in the Star Tribune (Minneapolis, Minnesota, USA) of Sunday 20^th April 2014 [page N3, column 1]:

Reba Clamp, of Minneapolis, opens the show as a woman in labor. By the end, it circles back to her final monologue, “Peed on a Stick,” which depicts an anxious woman, who has just embarked on a new career, waiting for the results of a pregnancy test. “She logically knows it’s not time to have a baby,” Clamp said, “but she really, really wants one.”

These are, in chronological order, the earliest occurrences of the phrase to pee (also to wee, to piss) on a stick that I have found:

1-: From Condom ads necessary message, by David Graupner, published in the Wisconsin State Journal (Madison, Wisconsin, USA) of Sunday 12^th February 1995 [page 3B, column 3]:

As I write this column it is Monday evening at about 7:45 p.m. The two girls and three boys in my household (all under the age of 14) are watching “Dave’s World” on the much-respected WISC-TV. Tonight’s plot involves birth control and contains such lines as “I’m peeing on a stick,” “Is that a vas deferens in your pocket or are you happy to see me?” and a discussion of tubal ligation vs. vasectomies.

2-: From Bridget Jones’s Diary, a novel by the British journalist, novelist and screenwriter Helen Fielding (born 1958)—as published in the Independent (London, England) of Tuesday 13^th June 1995 (here):

Friday 9 June
Went to the chemist to discreetly buy a pregnancy test. […]
[…]
For the first two hours this morning I kept staring at my handbag as if it was an unexploded bomb. At 12.30 I could stand it no longer, grabbed the handbag, got in the lift and went to the loos two floors down to avoid the risk of anyone I knew hearing suspicious rustling. For some reason the whole business suddenly made me furious with Daniel. It was his responsibility, too, and he wasn’t having to spend £8.95 and hide in the toilets trying to wee on a stick.

3-: From Make the Connection: Ten Steps to a Better Body—and a Better Life (New York: Hyperion, 1996), by the U.S. talk-show host Oprah Winfrey (born 1954) and the U.S. exercise physiologist and personal trainer Bob Greene (born 1958) [Oprah’s Story, by Oprah Winfrey: page 4]:

By the end of the year I was 150 pounds. Thus began diet after diet in my search for the perfect weight-loss method. First, the Atkins diet: Eat all the fat, cheese, butter, eggs, and bacon you want, just don’t eat any carbohydrates, and be sure to pee on a stick to make sure your body isn’t completely breaking down.

4-: From Watch out where you park: You might end up pregnant, about supermarket car parks, by Irene Haskins, published in the Columbia Daily Tribune (Columbia, Missouri, USA) of Thursday 22^nd January 1998 [page 8A, column 5]:

We’ve all seen perfectly able-bodied people park in handicapped spaces, so don’t think a few fatties won’t try this flim-flam. Also, don’t think a smattering of unscrupulous skinny Minnies won’t stoop to padding themselves. Pretending to be pregnant is a small price to pay for such a perk.
Should there be an in-house doctor to give exams? Will suspect parkers be forced to pee on a stick to see if it turns positive?

5-: From a review of the U.S. sitcom For Your Love—review by Tim Goodman, published in the San Francisco Examiner (San Francisco, California, USA) of Tuesday 17^th March 1998 [page B2, column 4]:

The lines are delivered with a paucity of comic timing. Peete, when told maybe she didn’t do the pregnancy test just right: “I think I know how to pee on a stick.”
She might as well have said this: “Here is a line I’ve been told to read.”