Is backpropogation used in convolutional neural networks?
Both Full Connected Neural Networks and Convolutional Neural Networks use backpropagation for training. 

What you said is right, both are feed forward neural networks, which means that the connections in the neural network start from left (input) and move towards right (output). But, it is feed forward only during inference time. During training i.e. when you are trying to find the weights/bias of the neural network for the task at hand, you have to use backpropagation. 

Training has both forward pass and backward pass.
Inference has only forward pass.

Also, when we say feed forward it means that there are no loops in the network. L1->L2->L3->L4 and so on. But, if you see recurrent nets, they have loops like L->L->L->L