I have just recently moved to Mass and want to know how long we need to wait before filing for a modification of Child Support due to the increased cost of living?

You'd have to check around for the local laws. Child support is usually based off what the father makes rather than the cost of living for where you are. If you need to make this change it will probably be after you have become a legal resident which can be around six months to a year.

I can say with almost certainty, that your child support will not be changed due to cost of living. You chose to move there.
Child support is based off a simple equation with both parents income being the factors, that is it.

I will agree, I doubt that your choice to increase your spending will be considered grounds for filing. In fact if you moved out of state with the children did you ask permission of the court to move first. If not the other party, if they were using their visitation rights, may ask the court to order you to move back.