Month() method can be used to get current month using apex in Salesforce. Also, format( ‘MMMM’ ) can be used to return the month in String since the Month() method returns integer.

Sample Code:

Integer intMonth = Date.Today().Month();
System.debug(
    'Current Month is ' + intMonth
);

To get the Month in String, use the following Code.

DateTime currentDT = System.now();
System.debug(
    'Current Month is ' + currentDT.format( 'MMMM' )
);