No. The two are quite separate, and may be provided for in different statutes or in different sections of the same statute, depending on the jurisdiction.
Exemplary damages, also called punitive damages, are intended to punish the offender (or to "make an example" of the offender). They are often awarded for wilful or particularly outrageous conduct. There is often wide discretion in whether to award them, and how much to award. They may have no clear relation to the amount or kind of damages done to the plaintiff.
The recovery of attorney’s fees, and sometimes other costs, is a significantly newer concept in tort law. The purpose is to be fair to a plaintiff who might have had significant expenses in establishing a valid tort claim. The further purpose is to avoid people with valid claims being deterred from proceeding on them because of such costs. It does not generally depend on any particularly outrageous conduct on the part of the defender.
However, in many states, the amount of fees awarded may be considerably less than the actual expenses incurred by the plaintiff. Particularly when the statute specifies "reasonable attorney’s fees", the court's notion of that is reasonable may be rather less than the market rate. On the other hand, some statutes in some states provide for "full costs" or "actual attorney’s fees" which will more fully compensate a winning plaintiff for all expenses involved.
All the above applies to all tors, not just defamation. In defamation cases, proof of actual malice is required in some US states for exemplary damages to be awarded. This concept was borrowed and significantly modified, in the famous case of New York Times vs Sullivan to require a Public Official (and later a public figure) to prove actual malice to win any recovery in a defamation case. The term is unfortunate, as the usual proof of actual malice does not establish any malice in the dictionary sense of the word.
In some cases there are also statutory damages, which are assessed without any proof of actual economic damage, either instead of or in addition to, actual damages (also known as compensatory damages). For example, US Copyright cases provide statutory damages if the copyright was registered before the infringement occurred, or within three months of publication, but not otherwise. The amount or range of statutory damages is specified in the law. Only some specific torts carry statutory damages.