The bites from bed bugs can cause symptoms that run from no symptoms at all to very severe reactions. It all depends on the personal immune system of the person who is fed upon and how violently that person's system decides to react to the presence of the foreign protein in their body. The protein, of course, is in the saliva of the feeding bed bug, and we may react just as we do to the feeding of fleas, mosquitoes, or ticks. Ticks, in fact, have a venom in their saliva that can cause a very severe "tick paralysis" in people, so who knows what other ingredients may be discovered in the saliva of bed bugs as they are more intensely studied. We might also compare this with the reactions people have to the sting of a bee or wasp, where that single sting could be life-threatening to some people but of little consequence to most others. 

Given this I would say that a fever is a "possible" reaction to a bed bug bite but definitely not a typical reaction. It also is important to note that as much as the topic has been studied there still does not appear to be any potential for The Common Bed Bug to transmit diseases to humans. They certainly are capable of sucking up pathogens from an infected person, but appear to have no mechanism for passing those pathogens along to another person. So, the fever would not be associated with disease transmission. This is an important message to give to customers who are able to read all sorts of nonsense on the internet. When Swine Flu hit the U.S. a couple of years ago the internet blabbered on and on about how blood feeding insects surely could spread the flu, which simply was not true. 

So, the symptoms of bed bug bites. At the bottom of the spectrum would be NO reaction at all. I have a friend who maintains a university colony of bed bugs by allowing them to feed on HIM, and he tells me that thousands of feeding sessions by bed bugs have never caused so much as a little itchy bump. The bite is essentially painless, so victims are not aware of the feeding unless they choose to watch. Most people might exhibit a small red welt that may itch for a few days. For others this welt may be much larger, become inflamed, and itch intensely. Reactions may even become worse over time as people continue to be bitten and their immune system objects more and more. The bite bump could become a blister, and scratching at this could open it, expose tissues underneath, and result in an infection that results in pus, pain, and maybe even that fever. 

It has been reported that some people may develop neurological disorders over time due to bed bug bites, and certainly the result of been fed upon at night in the sanctity of your own bed can lead to nervous behaviors, sleeplessness, agitation, and somewhat poor health due to the stress of anticipating more of the bugs in your bed. For some people this could make them ill and result in that fever again. A child may be less likely to obsess over this than we smart adults, so who knows if the child would develop the stress symptoms. However, there would seem to be other more likely reasons for a child to develop a fever during the winter months, such as the flu or even a cold. We could not rule out a reaction to bed bug bites, but I would avoid deciding these bugs must be present until you actually see the bugs or their direct evidence in that home.