Your ideas for your alarm clock class sound OK.
I wouldn't call it "Alarm_Clock" with an underscore in it; just "AlarmClock". You can call it whatever you want ofcourse, but there is a defacto coding standard / style that almost every Java programmer uses, in which class names are written in CamelCase, just like all the classes and interfaces in the standard Java API. See:
Code Conventions for the Java Programming Language.
Class Calendar and class Date are both useful for storing dates and times; you could use either. Class Calendar contains a number of useful methods for doing date arithmetic.
Instead of the AlarmClock class having separate properties for hour, minute, second etc. you could also just give it one Date property: