The simple answer is yes, but only because the question is so vague.
If by "4 different outputs" you mean four different output voltage levels at a single node, then this can be done with a 2-bit counter driving a simple D/A resistor ladder. This is my first guess at what you are asking about, based on the wording. If a bistable has two possible output voltages, then a tristable should have three ...
Another option is that you are talking about four different points in the circuit, four nodes or four output pins, that cycle through a pattern as the button is pressed repeatedly. In this case, a 4017 can be wired to cycle through anything from 1 to 9 outputs.
As above, a truth table of your required results is critical in answering this question.