Did Kolmogorov complexity influence the development of communication complexity?

The first link says that

communication complexity studies the amount of communication required to solve a problem when the problem is distributed between two or more parties.

Whilst the second link says that

Kolmogorov complexity of an object, such as a piece of text, is the length of the shortest computer program... that produces the object as output. It is a measure of the computational resource...

Had you read further down in the first link you would have discovered:

Note that unlike computational complexity theory, communication complexity is not concerned with the amount of computation performed by Alice or Bob, or the size of the memory used, as we assume nothing about the computational power of Alice or Bob.

So, no, the two concepts are not at all related other than sharing the word 'complexity' in their names. And that connection is spurious.