If you are not happy with your pay aren't you basically indirectly implying that you are not happy with your job?
I would say if you are not happy with your pay, find a new job or at least offers and take it from there.
You may not be unhappy at your current job (not your dream job, but not terrible either), so if you present any real offers to your current employer and they like you and see you as a valuable asset that they do not want to lose I would think they would offer you more if that's what you wanted. Then you can decide from that point if you want more pay (assuming they offer to pay more and keep you) or do you want more pay and a different job?
I would say if you are not happy with your pay, find a new job or at least offers and take it from there.
You may not be unhappy at your current job (not your dream job, but not terrible either), so if you present any real offers to your current employer and they like you and see you as a valuable asset that they do not want to lose I would think they would offer you more if that's what you wanted. Then you can decide from that point if you want more pay (assuming they offer to pay more and keep you) or do you want more pay and a different job?