If you are building a library, you should allow the user to choose how they want to handle it either by having methods that return the first/last full/partial week of the month or by passing an argument that indicates whether you want the full/partial week. Clearly document whichever is the default if you provide methods that don't require the specification.
public Week GetFirstFullWeekOfMonth( DateTime forDate ) { }
public Week GetFirstPartialWeekOfMonth( DateTime forDate ) { }
Or
public Week GetFirstWeekOfMonth( DateTime forDate, bool includePartial ) {}