I need to update the password for a user that is used in possibly multiple health monitors across multiple partitions.

Anyway to search all monitors across multiple partitions for a user in health monitors?

No problem.  The way I do this is say I need to search for a username svc_sp_F5MONITOR_M and ignore case, I'd run the following command which will give you the ltm monitor and the username so now you know which monitors you need to update.

tmsh -c "cd /;list ltm monitor recursive" | egrep 'ltm|username' | grep -B1 -i F5MONITOR_M

Following the help for altering monitor options as shown here

Create/Modify
    create https [name]
    modify https [name]
        app-service [[string] | none]
        cert [ [cert list] | none]
        cipherlist [string]
        compatibility [enabled | disabled]
        defaults-from [name]
        description [string]
        destination [ip address][port]
        interval [integer]
        key [ [key] | none]
        manual-resume [enabled | disabled]
        password [none | [password] ]
        recv [none | [string] ]
        recv-disable [none | [string] ]
        reverse [enabled | disabled]
        send [none | [string] ]
        time-until-up [integer]
        timeout [integer]
        transparent [enabled | disabled]
        up-interval [integer]
        username [ [name] | none]

So the command to change the password for a monitor would be

tmsh modify ltm monitor https /Integration/https.web-mod.int.thezah.com password MyNewPASSwoRD