The expression "Morton's Fork" originates from a policy of tax collection devised by John Morton, who was Lord Chancellor of England in 1487, under the rule of King Henry VII.
Morton's approach was that if the subject lived in luxury and had clearly spent a lot of money on himself, he obviously had sufficient income to spare for the king. Alternatively, if the subject lived frugally, and showed no sign of being wealthy, he must have substantial savings and could therefore afford to give it to the king. These arguments were the two prongs of the fork and regardless of whether the subject was rich or poor, he did not have a favourable choice.
The phrase is rarely used these days as there is a more common analogy when someone has a dilemma and has to choose between two equally unpleasant alternatives. We tend to say that they are either "Between the Devil and the Deep Blue Sea" or "Between a rock and a hard place".