There is no direct relationship between these two concepts. However we can find some indirect ones.
According to Merriam Webster,
kernel means a central or essential part
which hints why they are called "kernel". Specifically, deciding "how to measure point-point similarity (a.k.a. kernel function)" is the central part of kernel methods, and deciding "what array, matrix, or tensor (a.k.a. kernel matrix) to be convoluted with a data point" is the central part of convolutional neural networks.
A kernel function receives two data points, implicitly maps them into a higher (possibly infinite) dimension, then calculates their inner product.
A kernel matrix (or array, or tensor) is convoluted with one data point to map the data point explicitly into an often lower dimension. Here, we are ignoring a subtle difference between filter and kernel (a filter is composed of one kernel per channel).
Therefore, these two concepts are indirectly related based on mapping to a new representation. However,
- Kernel functions map implicitly, but kernel matrices map explicitly,
- Kernel functions cannot be stacked over each other (shallow representation), but kernel matrices can be since the input and output (explicit representations) has the same structure (deep representation),
- The non-linearity of map is integrated into kernel functions, but for kernel matrices, we should apply a non-linear activation function after the (input, kernel) convolution to reach a similar non-linearity,
- Implicit representations cannot be learned for kernel functions, a specific function implies a specific representation. However, for kernel matrices, representations can be learned by adjusting (learning) the weights of kernels, and can also be enriched by stacking kernels over each other.