MySQL time in shared hosting: is it stable / possible to change?

I have a question regarding shared hosting: is it possible to change the MySQL time from cpanel?

If not, do shared hosts often change the timestamp on their shared server? For example, I was surprised that HawkHost shared would randomly change their MySQL time in 1 hour (that is, a week ago the difference between the PHP timestamp and the MySQL time was -5 hours, but nowadays it is -6). I just want to make sure that if I choose a shared host, the MySQL time will never change.