Date range validator
What jumps me most about this code is that all you do is in 2 functions.

consider moving your "parsing" operations to separate functions, one for validating the format required for parsing and one for the actual parsing:

function isValidQuarterInput(datestring){
   return datestring.split(" ").length == 2;
}

function parseQuarterInputToMonth(datestring){
    var mySplit = datestring.split(" ");
    //move your Q1,Q2,... mapping here
}



then in check_date_period() you can just call the following

if(isValidQuarterInput(from)){monthFrom = parseQuarterInputToMonth(from);}
// this is the alternative ternary operator
monthTo = isValidQuarterInput(to) ? parseQuarterInputToMonth(to) : monthTo;


your var result = false is unnecessary, as you never use it elsewhere, you can just do the following for return: 

return dateOne &lt; dateTwo;



I personally would set a boolean wasParsed to be set to true on correct parsing of one input to jump to the new Date() part. This can potentially save execution time, but that's minor. You would have to move the parsing in the following order though:

var wasParsed = false;
if(from.length==4){
  yearFrom = from; monthFrom = 1; dayFrom = 1;
  wasParsed = true;
}

if(!wasParsed){
   if(isValidQuarterInput(from)){
      wasParsed = true;
      monthFrom = parseQuarterInputToMonth(from);
      yearFrom = from.split(" ")[0];
      dayFrom = 1;
   }
}

if(!wasParsed){
  //here is the full date parsing
}
var dateOne = new Date(yearFrom, monthFrom, dayFrom); 
wasParsed = false;
//repeat with to