Is there a way of how I can change the hosting server time-zone settings from my hosting portal? Or do I need to request GoDaddy to merge my current hosting account from US server to an European server which has its server time zone as (UTC+-01:00) Amsterdam, Berlin, Bern, Rome, Stockholm, Vienna?
What do you mean by "hosting server time-zone settings"? How do you access them? What language are you using? Are you doing that in the server or in the client? Are you also getting time and date for somewhere else, such as a database? Are you getting time and/or date from the client and processing in the server?
Unfortunately without clarification, no one can help.