The short answer has to be, "Yes." The connection between diet and nutrition and its effects on health in humans is well documented by now, and general studies have been done in regard to pets. It's pretty well known that the body's ability to function normally is closely related to proper nutrition.
Since behavior depends on how well all systems perform, it should be an obvious conclusion. For example, normal brain development depends on sufficient levels of polyunsaturated fats, especially the omega 3 and omega 6 fatty acids, which can affect cognitive performance, and thus, behavior. In addition, antioxidants may help decrease cognitive decline in older individuals, and tryptophan could be related to decreased aggression or fearfulness, especially in dogs.
However, fewer studies have been done with pets than with humans. Thus, the relationship of specific nutrients to specific behaviors is not well understood.
Throughout human history, foods have been assigned certain qualities, and eating them supposedly confers upon one that quality. The most notable example might be aphrodisiac foods. However, it appears that a belief in an expected outcome is the important aspect of a particular result, rather than an actual, provable connection between physiology and behavior.
Since pets don't have belief systems, other than believing their owner will act in a certain way, connected with a certain circumstance (such as feeding them every day at 5 p.m.), their behavior cannot strictly be attributed to specific foods from a psychological perspective without considering stimulus-response training.
Most pets can be conditioned to behave according to a set of stimuli that we provide, such as eating at specific times, as noted, but they also are able to learn preferences for foods by experience. Sometimes they are conditioned negatively to avoid some things associated with an uncomfortable result, such as punishment, and conversely, can learn to desire some things because it has a positive result, such as a tasty treat for behaving well or performing a trick.
The bottom line for our pets, as well as for ourselves, is to provide a healthful and balanced diet for optimal health, and thus, feelings of well being that can lead to positive behavior.