Yes you can ride a motorcycle up the Pikes Peak mountain if you want too as I've seen many motorcycles driving up and down the Pikes Peak mountain road.
You'll have some beautiful views from Pikes Peak mountain and it would be a lovely motorcycle ride.
Be sure to stop and take some pictures for lasting memories of Pikes Peak and your drive through Pikes Peak as it's very beautiful.
I drove up Pikes Peak in my car a few years ago but I did see quite a few motorcycles driving up Pikes Peak as well.
The drive through Pikes Peak is around 19 miles and takes between 30 minutes to an hour depending on how often you stop and explore and rest.