1) I have a suspicion that she is not telling the truth.
-You have a feeling that she is not telling the truth.
2) I have a suspect that she is not telling the truth - not correct
I should be either "I suspect that she is not telling the truth" or "I have a suspect that confirms that she is not telling the truth"
3) I suspect that she is not telling the truth
You have a feeling that she is not telling the truth
4) I have a suspect that confirms that she is not telling the truth
-a person who you believe to be guilty of a crime (a suspect) can confirm that she is lying