How can I gather the visitor's time zone information in GMT offset hours?

Ans:

     var offset = new Date().getTimezoneOffset();

The time-zone offset is the difference, in minutes, between UTC and local time. Note that this means that the offset is positive if the local timezone is behind UTC and negative if it is ahead. For example, if your time zone is UTC+10 (Australian Eastern Standard Time), -600 will be returned. Daylight savings time prevents this value from being a constant even for a given locale