This has been my only last-minute booking so far which made me a bit nervous since I had to do so much research on accommodations before hand and knew that just because a website says an apartment or hostel is pet friendly doesn't necessarily mean that the information is accurate. Having no other place to stay, I booked at Sleephere on Hostelworld and crossed my fingers.
When I arrived, the owner was definitely surprised that I had a dog with me and was a bit nervous about booking me in at first but instantly fell in love with Olive and made an exception. He booked us in a private room at no extra cost (as it wasn't booked) and even had a dog of his own roaming around.
I cannot recommend this hostel / guest house enough. It's very clean, the owner and workers take great care of you and you feel more like you're living in Brussels than simply staying in a hostel.
Olive was even allowed to roam free without a leash and play with the owners dog.
To say I lucked out this time would be an understatement, as I know not everyone will be this understanding and generous. So I wouldn't say to just book and hope for the best because you could end up having to make a last minute booking at more expensive accommodation which could take days of your backpacking budget straight from your pocket.
For those of you hoping to visit Brussels with your furry friend, stay at sleephere. You're just 10 minutes from the main square (Grand Place) and your dog will be more than welcome here, they'll be loved. Just contact the owner ahead of time and tell him your situation. As long as the private room isn't booked up and you keep him informed, there's no doubt in my mind that he'll be supportive.