HikeArizona.COM

I made a loop/shuttle of some of these trails last year, it could be the area that you are referring to/considering ( I did not search on alltrails for what you speak). Even if it not one-and-the-same, it's a fun area with springs and a couple of small peaks to bag. There are myriad trails back in there to keep you occupied - more than what you read about on summitpost.
http://www.summitpost.org/bear-canyon/697264

Starting point (move around on the map to the east and north to see all the options).....>> https://hikearizona.com/map.php?QX=6330

Bear CREEK runs along the highway and generally has some water in that area, sometimes a LOT. Bear CANYON (not 'trail') heads up ENE away from the highway just above there. The trail along Bear Creek is called Green Mountain (don't ask me why, as it goes AROUND Green Mountain).

@swingdance
I don't think it exists. That user posted it seven years ago and nobody has posted a hike there since then? Seems pretty dubious for something that is adjacent to the Catalina Highway!

Nonetheless, if you don't mind off-trail scrambling, you can probably follow the route he posted if you want. Follow the drainage downstream from the picnic area as far as you want and then turn back. Just don't expect there to be a maintained trail, and hopefully you don't mind the sound of traffic since it is never more than 100 feet from the highway.

You might want to hike upstream in Bear Canyon from Middle Bear picnic area (or start at General Hitchcock) on the https://hikearizona.com/decoder.php?ZTN=771 (This is also what hikeaz posted above).

Some of the Tucson users might be able to confirm that it doesn't exist, but I think it's just one more example of AllTrails being wrong.

@swingdance
I took a look at the route and its just the social trail that starts at lower green mountain/bug spring trailhead and goes through the picnic areas downstream. So many better options for hiking in that area and a mistake to name it "Bear Canyon Trail".