What exactly is the linear layer in the transformer model?
In the original Transformer article, these linear layers are just matrix multiplications.
As described in the paragraph you referred to in your question $W^Q$ is a matrix of dimensions $d_{model} \times d_k$, that is, it is a fully connected layer with $d_k$ units.
In practical implementations, these have the optional addition of a bias vector. You can see their actual definition in the fairseq code.