Conflict of interest?

A conflict of interest in its common meaning is that two, or more, parties are involved in some sort of cooperative venture that is intended for the common benefit of all parties, but there is a benefit to one of the parties that could, or does, affect the decision making of that party to the detriment of the others.

The one party tries to serve two masters (self and group) at the same time, and that is not possible when the two masters have separate needs.  That is the conflict.  The "masters" are the "interests" in the phrase.

There is a conflict in your example.  It is a conflict of desires, though, rather than of "Interests".  They are each serving themselves.  There is no pretense of acting to the benefit of the other, therefore no conflict of interests.

No, that's not a conflict of interest.  

A conflict of interest occurs when someone in a position to make a decision for others decides in his/her own favor.