No, they were independent developments.
However, the situation is a little bit more complex. Hurd is not just a Linux with a different kernel, also the structure of the OS is highly different.
GNU Hurd is a microkernel. That means that the actual kernel does as little as possible. The ordinary kernel functionality is being done by user space processes, communicating with the microkernel and with each other.
For example, the ext4 filesystem driver is a kernel module in Linux. That means it is a collection of functions in a library, what converts the ext4 filesystem operations to block device operations.
In Hurd, the ext21 filesystem driver is essentially a daemon, service. Just like, for example, the Apache web server. The ext4 filesystem operations are talking with this daemon.
A Microkernel-based OS has also the feature, that the actual microkernel is actually an easily replaceable part of the system. The important part of the system is the collection of its daemons.
GNU Hurd is the abbreviation of "Hird of Unix Replacement Daemons". During its development, the actually used microkernel was changed multiple times.
Thus, what we understand on the "Linux kernel", is a "microkernel + the collection of the Unix replacement daemons" in the sense of the GNU Hurd.
- Some of the code of the daemons is probably derived from Linux kernel code. Particularly the filesystem drivers.
- Its microkernel is an entirely independent development from the Linux kernel.
The sum is that probably it has much Linux kernel code, it is mainly a different product (note, both systems being GPL, it is not a major question from the intellectual property view).
1As far I know, ext3/4 support is not developed yet.