Yes, it is possible to use convolutional layers in a reinforcement learning (RL) agent approximation function for action values (e.g. Q learning) or for policies (e.g. REINFORFCE).
In fact, any learning system capable of online learning of functions from example inputs and outputs will work with RL. The RL component will generate the examples to learn by taking actions in the environment or in simulation, and calculating some value such as the expected return. These examples are drawn from different distributions as the agent becomes better at the task, which is why online learning is important - the agent must forget the values associated with earlier experiences and replace them with new values as it improves its performance. Neural networks work for online learning by default, unless you make changes to them to prevent that.
That means you are not restricted to simple feed-forward networks. You can use CNNs, RNNs and other flavours of neural network provided you design them to output your value function or policy. Which will be best to use depends on the nature of the environment and your input signals. CNNs are a good choice whenever there is a structured arrangement of similar inputs - that includes image data, also many board games.
If you have not already, you may want to get hold of a (free PDF) copy of Reinforcement Learning: An Introduction. In chapter 16, section 16.5 the authors explain the original DQN project which learned how to play video games, including a discussion of the neural network architecture and pre-processing used. This is nowadays a well-known result in the RL community, you will find discussions, examples and implementations of it in many places.
One of the original researchers on DQN, David Silver, has published a lecture series on RL, with videos available on YouTube. He is also associated with DeepMind's Alpha Go project, which is another example RL system that uses CNN architecture internally.