Someone who stubbornly refuses to talk
Solution 1:
Sulking, "expressing ill humor or offense by remaining sullenly silent or withdrawn" is one such word, and to some extent, pouting, "to be or pretend to be ill-tempered; sulk".
Solution 2:
If I understand you correctly, the word is "sulking". This means to refuse to speak to someone because you are unhappy with them. It is normally used when talking about children. If you say an adult is sulking, you are saying that they are not only not speaking but are acting in a childish manner.