Maybe, Probably, Possibly? This pictures looks like the old thermostat is using Y, W, G, Rh and Rc terminals (Rh and Rc are jumpered). The Compatibility Checker will probably tell you it will work but in some cases there will not be enough power. However, it is, by far, better to have a C (common) wire. Sometimes Nest Thermostats work without C, but sometimes not.
It is possible there is a 5th wire in the cable that could be used as a C (common). It would also need to be connected at the HVAC system. There are also adapters and transformers that can provide C (common) but it is best to have it come from the HVAC system.
