Is it acceptable to pass an Exception into a function that then may or may not throw it?
As Winston said, use full names for your variables.  Also keep the case consistent - then when you write $Object instead of $object it will be more obvious.

Static should almost never be used if you are writing object oriented code.  You basically have namespaced functions with this code.  Rather than write a static utils class for SQL I would write an SQL class (or use an existing one).  If you want objects of a class to be created PDO can do this for you.

As far as the exception goes:

No, you shouldn't do that.  Your code creates an object that it doesn't need to every time the function is called.  At the worst you would have passed in the exception string and instantiated with new Exception in fetchIdIntoInstance.

Main Recommendation

You are better of dealing with an exception after it has happened at a level where you know why it happened.

Keep your lower level functions simple:  If something strange happens throw an exception.  In your case I would assume that your sql should work in fetchIdIntoInstance and throw an exception if you can't make the new instance.

At a higher level database operations will have a meaning.  You will be doing something specific that is easily understood.  Maybe you will be adding a record to a table or reading an object that you expect to be in the database.  At this point you can handle the exception and write meaningful information to a log file or something similar.

You could catch exceptions in getThatOneInstance but it does not seem like the right place.  I think you should catch it higher up where you know what to do with it.