Not necessarily. Coaching is focused on acquisition of skills or knowledge and is most often performance based. Coaching is designed for achieving a certain result: namely, the increased competency of a given coachee. A coach, therefore, needs to be an expert in the area s/he is coaching and is accountable for results. The skills necessary to be a good coach include one's ability to share one's expertise with the coachee and to ensure that person develops the skill sets involved.
A mentor is focused on developing a relationship based upon trust which will allow the mentoree to share the issues that impact his/her success. Therefore, to be a good mentor means you need to be good at creating relationships. You may or may not be an expert in the areas a mentoree wishes to develop but you do need to be able to create that climate of trust.
So being a good coach doesn't mean you are going to be a good mentor. The skill sets are different for each. Both are valuable but you want to ensure that the right solution is applied.