No, probably not -- unless you happen to get lucky and the kernel weights you started with just happen to be good ones. But if you choose the kernel weights randomly, your procedure will probably work very poorly.
You can always try it out yourself and see what happens.
That said, you might be thinking of "fine-tuning". In fine-tuning, we first train a model for some image recognition task, on some training set. This stage involves learning both the kernel weighted and fully-connected layer weights. Now, suppose we have a new task (hopefully similar to the original task, but a bit different) and a new training set. Then you can take the prior model, hold the kernel weights fixed, and try doing the learning procedure (backpropagation etc.) on the new training set, adjusting only the last few fully-connected layers. Sometimes this is highly effective: the kernels in the first few layers that are useful for one image classification task are often still useful for other image classification tasks. For instance, people sometimes take a good model trained on ImageNet for general objection recognition and then use it with fine-tuning to do some other similar but not identical image classification task. This potentially saves training time and requires less training data, and can sometimes work well.