Yes, but it would be tricky.
First, I will assume from your question that this bike riding will happen indoors, in a shirt-sleeves environment that does not yet exist, such as a lunar station. Riding a bike during EVA would be a very different proposition, and I will not comment on it.
Several lunar station proposals have included stationary bicycles as a possible exercise equipment. Second, I will assume that the question is not about stationary bicycles, but about actually moving around inside the pressurized facility on a bike.
1. Level surface
- Traction is the product of normal force and the traction coefficient of the contacting materials (see Traction wikipedia). Normal force on a level surface is the combined weight of the bike and the cyclist, which is 1/6 on the Moon. Unless extra weight is added, or the material of the tires and the floor are made more adhesive, the bike tires will slip more easily than they would on Earth.
- Acceleration and deceleration must be slower, more gentle than on Earth, to keep both wheels in contact with the ground. To stay balanced, the net force direction from the center of mass must stay between the contact points of the two wheels against the floor. When moving on a level surface, the net force is a combination of the horizontal force and gravity. Since gravity is smaller, the net force angle is dominated by the horizontal accelerations. If the center of mass (of the whole bike+bicyclist system) is not lowered, it is easier to accidentally pop a wheelie in the back or front when either braking or accelerating, and lose control of the bike.
- The Leaning angle when turning must be steeper than on Earth at the same speed. This is also due to the vertical component of the net force being six times smaller. To stay balanced, the net force angle from the center of mass must point inside the narrow base formed between the contacts of the two wheels against the ground. Di Prampero, 2000
The difference in the leaning angle is illustrated in this picture:
2. Concave surface
Depending on architecture, there might be cylindrical sections in a lunar base that could be reserved for exercise, in the manner of a silodrome or "Wall of Death". A 8-10 m diameter cylindrical section would present a continuous circular track for either biking or running. Using a bicycle it would be easier to maintain a steady speed, and generate 1 g of centrifugal force against the wall. Again, starting and stopping would seem to be the most difficult stages. After getting up to speed, the centrifugal force will help with traction as well as balance, since it is pointing "down" from the center of mass towards the concave wall.
On a bike, the circular track may work even if the cylinder is horizontal, and the bike passes a point where it is upside down. This is of course similar to the "Globe of Death" feat performed in many circuses on Earth, but it will be easier to perform because there is less gravity to overcome. In the vertical loop, difference in net force between the top and the bottom of the cylinder is only 1/3 g, either adding or subtracting 1/6 g to the centrifugal 1 g.