To add to Archimedes comments, it is difficult for anyone (man or woman) to add a lot of muscle mass while running a calorie deficit to lose weight, as the body tends to burn protein for energy, rather than creating new muscle tissue.
Also, fat is considerably bulkier than muscle. Take a look at the photo at www.dailyspark.com/blog.asp?post=quiz_how_
Bulk in the belly is because of fat stored there, not muscle.
In any event, the abs are a long thin sheet of muscle. Even if over a period of say 12 months, you were able to add 50% to your abdominal muscle mass, because it is a thin sheet, you aren't going to be adding noticeably to your waist measurement.
A strong core is important for posture and balance, and actually makes other forms of exercise easier. And as Archimedes said, strength training is a significant fat burner. You SHOULD definitely be including core exercises in your overall strength training program. Although it should be noted that burning fat is an all-body process, and core exercises won't specifically help you target body fat.
Your trainer is wrong in her concerns, and wrong in her advice on structuring an exercise program.