FileZilla: Can I run a remote shell command via FileZilla client?

Some, but not many, FTP servers support SITE EXEC command. You can enter that in FileZilla using Server > Enter custom command. E.g.:

SITE EXEC tar -zxvf myarchive.tar.gz

Other than FTP, FileZilla supports only SFTP protocol, which does not have any way to execute command on the remote side.

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Though it is common that if you have an SFTP access, you also have an SSH terminal access. Some SFTP clients (but not FileZilla) have a built-in terminal feature that allows you to execute commands over an SSH terminal session that seamlessly opens using the same credentials you used for SFTP.

For example in WinSCP, go to Commands > Open Terminal:

You can even save the command and easily reuse it.

With FTP protocol, you can execute the SITE ... command from the same window.

(I'm the author of WinSCP)

As this link points out, the answer to the question is still no. For executing remote shell commands use an ssh client.