I can't tell you why you get no results with a Google transit search, but I can tell you that you won't find anything useful searching the OC Transpo web site unless you already know exactly where you're going, and it won't show you what you need to know anyway since OC Transpo doesn't really go into Gatineau.
So, here's the kind of so-so news.
There are buses which reach the southermost edges of the park boundaries, but they do not go into or around the park, and do not reach as far as the visitor centre or many of the places within the park you might want to go. You'll have a bit of a hike ahead of you. Fortunately there are plenty of hiking trails...
So, Société de transport de l'Outaouais routes 27, 35 and 48 reach the southeast and southwest boundaries of the park respectively, and you can get either of these in downtown Ottawa. They enter Wellington Street at the Portage Bridge and travel to their last Ottawa stop at Rideaux and Cumberland, before returning to Gatineau via King Edward Avenue. Depending on the route you take, expect 40-45 minutes between Ottawa and the park boundary.
The STO system map shows a few other routes which overlap those at some points, in case those might also be useful to you. (For instance, routes 35 and 48 only run Monday-Friday, and you may want to find a weekend bus.) Cash fare is $3.90, and some discounts are available.